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1
Rinse peas well in a strainer or colander.
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2
Melt the butter in a large Dutch oven or soup pot, and saute onions, carrots and celery over medium heat until lightly browned. Sprinkle 1/4 teaspoon salt over all.
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3
Add the crushed garlic and saute until fragrant, about 1 minute.
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4
Keeping heat the same, add the broth and peas. Scrape bottom of pan to loosen any browned bits and stir to mix everything up.
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5
Trim all visible fat from the ham and add ham to the pot. It's fine if it breaks into a few chunks, but they need to be large enough to pick out later.
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6
When soup comes to a light boil, use a wad of paper towels to dab any whitish foam off the surface.
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7
Lower heat to a simmer, cover, and cook for 2-3 hours. About an hour in, use tongs to pull out all the ham and either chop into bite-sized pieces or snip with kitchen scissors right over the pot. Now is also a great time to taste for seasoning; you may need to add nothing at all, but see Ingredients for ideas.
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8
If you like, blend some or all of the soup with a regular blender (more trouble) or immersion blender (zero trouble).