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For the soup:
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Place chicken and giblets in a close-fitting 5-quart soup pot, preferably of enabled cast-iron and add up to 10 cups of water, which should cover the chicken.
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Bring to a boil, partly covered, then reduce to a simmer and skim foam as it rises to the surface.
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When foam subsides, add the whole celery stalk, carrot, whole onion, parsley, and peppercorns.
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Keep at a low simmer for about 3 hours, stirring every 30 minutes, until chicken begins to fall from bones.
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Replenish water if chicken is not covered during the first 2 hours of cooking.
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While chicken is cooking, prepare split peas.
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Wash them in 2 changes of cold water and pick over to remove stones or discolored peas.
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Place in saucepan with about 3 to 4 cups water and boil for 2 or 3 minutes.
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Drain off water and rinse peas.
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Add about 3 to 4 cups fresh water and bring to a boil and cover; let stand for about 30 minutes.
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Remove chicken soup and discard giblets, vegetables, and peppercorns.
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If you plan to serve chicken meat in soup, remove skin, bones, and cartilage and break breast and thigh meat into spoonable pieces.
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Strain soup through a sieve and skim off fat.
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Return to rinsed soup pot and add soaked, drained split pea.
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Simmer gently, partly covered, for about 20 minutes while preparing the other vegetables.
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Heat butter and oil in a 10-inch skillet.
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Add chopped onion, leeks, and diced celery and saute gently, stirring frequently, for 8 to 10 minutes or until vegetables soften but are not brown.
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Add to soup with string beans and simmer gently, partly covered, for about 10 minutes.
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Wash zucchini well, trim off ends, and cut into vertical quarters.
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Using a small paring knife or a spoon, scoop out inner pulp with seeds.
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Slice remaining zucchini shell into 3/4 -inch pieces.
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Cut cabbage into quarters, wash under running cold water, and trim out white core.
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Shred cabbage as for coleslaw, eliminating tough white ribs.
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Add zucchini and cabbage to soup and simmer gently for another 20 minutes.
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If you use Swiss chard, trim off all white stems and veins.
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Stems can be cooked as a separate vegetable another day.
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Wash leaves well and chop or slice coarsely.
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You should have 3 to 4 cups.
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Place in saucepan with enough water to cover and parboil for about 3 minutes or until leaves begin to wilt.
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Drain well and add leaves to soup, discarding their cooking water.
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The soup can be prepared up to this point and stored, covered, in the refrigerator for up to 24 hours.
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Bring to room temperature before reheating.
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About 20 minutes before serving, add rice or potatoes to soup and cook until tender.
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Season with salt and pepper, but do not oversalt as pesto will be added.
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Serve in heated bowls, with or without chicken.
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Pass Pesto Sauce at the table.
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I prefer this without extra cheese sprinkled on top, but it can be passed at the table.
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Leftover minestrone can also be stored, covered, in the refrigerator for up to 24 hours or frozen for 6 weeks, but rice will expand considerably so soup will need thinning with water or broth in the reheating.
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Potatoes may disintegrate a bit, but that can be pleasantly soothing.
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For the pesto:
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Cut garlic cloves in half vertically and remove and discard inner green-tipped sprout.
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Cut garlic into small pieces, and, using a blender or mortar and pestle, crush to a paste with coarse salt.
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You can also crush garlic through a press and then mix with salt.
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Slice basil leaves and add to garlic.
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If using a mortar and pestle, trickle oil in slowly, working it in as for mayonnaise.
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If you are using a blender, add 1/3 cup oil and blend at moderate speed until you have a smooth emulsion, adding the rest gradually.
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If you add cheese, stir it in by hand.
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Add salt as needed.
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Extra pesto can be spooned into a narrow, small bowl or jar and topped with 1/2 -inch layer of olive oil.
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Cover and store in refrigerator for up to 1 week or in freezer for up to 1 month.