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1
Cook onions in oil in a large, deep saucepan, stirring occasionally, until onions are very soft and a deep brown color, 25-30 minutes. As the onions cook down, they may start to stick to the bottom of the pan; whenever this happens, deglaze with a small quantity of cold water, scraping with wooden spoon to dissolve brown bits.
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2
Remove about 1/4 of onions from pan, reserving for garnish. Season with salt and pepper and cover to keep warm.
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3
Sprinkle flour over the onions in the pot, stirring to incorporate. Add beef stock and dried thyme and bring to a boil. Lower to a simmer, adjust seasoning, and add pasta. Cover pan and simmer, depending on type of pasta, for 6-9 minutes, or until pasta is al dente and most of the liquid is absorbed. Stir once or twice during cooking to prevent sticking.
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4
Remove pasta from heat and stir in shredded Parmesan. Adjust seasoning if necessary. To serve, top with reserved onions and Parmesan crisp.
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5
Heat a nonstick pan over medium-low heat. Sprinkle a small handful (about 1 1/2 tablespoons) of grated Parmesan into the pan to form a circle about 3 inches in diameter. Cook until cheese is somewhat melted and beginning to take on color, 1-2 minutes. Using a spatula, gently loosen the cheese and flip, cooking for an additional 30 seconds.
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6
Place on paper towels to cool, or, for a curved shape, immediately drape over a rolling pin or other cylindrical surface.
