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First, the pasta: Mix the flour, puree, salt, and olive oil together until homogenous.
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Now, teaspoon by teaspoon, add the water, mixing after each teaspoon, until it starts to bind into a firm, dry dough.
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A wet and sticky dough is your enemy.
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Don't make your pasta dough your enemy; mix only as much water as you need.
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It's just shy of a half cup, probably.
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Really mix the dough aggressively to get the gluten and elasticity going.
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Now, you could roll your dough out by hand with a rolling pin until really thin, and then slice into fettucine.
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However, I recommend using a pasta machine, rolling to 7, and then using the fettucine attachment.
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Either way, hang it out to dry a bit.
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I like using a laundry rack.
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So, the sauce... heat the olive oil on low in a 3 quart saucepan, and then toss in the onions.
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Let the onions get a head start for a few minutes, and then add the garlic, pepper flakes, salt, and basil.
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When the onions have completely given up the ghost and are translucent, and not a moment sooner or later, add the puree and mix.
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Turn the heat to medium, and let's start loosening the sauce up.
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Add the milk, mix, and then add the water.
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Set a large vat on the stovetop and boil enough water to cook 5 cups worth of noodles.
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Simmer the sauce for about 10 or so minutes, and incorporate the cheese.
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Add salt to taste, simmer for another little while on low while you cook the pasta and until you are sure there isn't a single bit of unsoftened rind specks in your sauce (disregard that bit about rind specks if you bought your puree).
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Toss your noodles into the vat of boiling water and cook until softened, likely just a few minutes.
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Think more like you are poaching an egg than boiling a potato in terms of timing for the noodles.
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Drain the noodles in a colander and run under cool water to stop the cooking process.
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Set the colander of noodles in a bath of warm water.
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In this way, you can take the colander out to scoop the drained noodles without worrying about cold or stuck together noodles.
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So, to serve, put some pasta on a plate, and then ladle some sauce in the middle.
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Enjoy.