Prep: grate the cheeses and set both cheeses in a bowl in the refrigerator so they stay cold (cold cheese melts more smoothly into the warm liquid). Dice the onions, mince the garlic, and measure the pasta. Having everything mise en place makes the cook faster and prevents overcooking the pasta.

Brown the beef: Heat a large heavy-bottomed pot (6–8 quart) over medium-high heat. Add 1 tablespoon olive oil, then the ground beef. Season lightly with salt and pepper. Brown the beef, breaking it into small pieces with a spatula until it’s just caramelized (about 6–8 minutes). Browning develops Maillard flavors that give the broth depth—don’t crowd the pan or you’ll steam the meat instead of browning it. Transfer the beef to a bowl and set aside, leaving any rendered fat in the pot.

Caramelize the onions: Reduce heat to medium. Add remaining 1 tablespoon olive oil and the butter to the pot. When butter foams, add the diced onions and a pinch of salt. Cook, stirring occasionally, for 12–15 minutes until the onions are golden and sweet. Slow, patient cooking of the onions builds natural sweetness and a rounded base flavor for the soup.

Add garlic and thyme: Add the minced garlic and dried thyme to the onions and cook for 30–45 seconds until fragrant. Garlic cooks quickly; brief cooking releases its aromatics without becoming bitter.

Make a light roux: Sprinkle the flour over the onions and garlic and stir for 1–2 minutes, cooking the raw flour taste out and giving the soup body. Cooking the flour briefly also prevents a starchy, grainy texture when liquid is added.

Deglaze and build the broth: Slowly pour in 1 cup of the beef broth while stirring to lift the browned bits (fond) from the bottom—this is flavor. Stir until the mixture is smooth, then add the remaining beef broth and the milk. Return the browned beef to the pot. Add the bay leaf and Worcestershire sauce. Bring the pot to a gentle simmer (not a rolling boil) over medium heat.

Simmer to develop flavor: Reduce heat to medium-low and simmer gently for 10 minutes to let the flavors marry and the roux thicken the liquid slightly. A gentle simmer concentrates flavor without breaking the milk (high heat can cause the dairy to scorch or separate).

Cook the pasta in the soup: Increase heat to medium, add the dry pasta, and cook for the time recommended on the package minus 1 minute so the pasta finishes cooking in the hot soup later. Cooking pasta in the broth allows it to absorb savory flavors; removing it just shy of done prevents it from getting mushy as it sits.

Finish with cheese: Reduce heat to low. Stir in the grated cheddar and Parmigiano in small handfuls, stirring continuously until fully melted and the soup is silky. Add the optional teaspoon of lemon juice to brighten the cheesy richness. Low heat and small additions prevent the cheese from clumping; stirring slowly yields a smooth emulsion.

Adjust seasoning and rest: Taste and adjust salt and pepper. Remove and discard the bay leaf. Let the soup rest off-heat for 3 minutes—the carryover heat will finish cooking the pasta and mellow the flavors so they harmonize.

Serve: Ladle the soup into warmed bowls and garnish with chopped parsley and an extra grating of Parmigiano if desired. A sprinkling of black pepper and a drizzle of olive oil (optional) adds a final glossy sheen and aroma.

Storage note: Cool to room temperature within 2 hours, store in airtight containers in the refrigerator for up to 3 days. Reheat gently over low heat, adding a splash of milk or broth to loosen the texture if the soup thickens.