Prepare the produce: peel and finely dice the onions, thinly slice the garlic, peel and dice the carrots to about 5 mm dice, and core and chop the tomatoes into roughly 1–1.5 cm pieces. Why: uniform dice on aromatics and vegetables ensures even cooking and a harmonious texture in the finished soup.

Heat a large heavy pot over medium heat and add the olive oil. Once shimmering, add the butter. Why: combining oil and butter gives the butter higher smoke resistance while still delivering that silky, nutty finish from the butter.

Add the diced onions and a pinch of salt. Cook, stirring occasionally, for 6–8 minutes until the onions are translucent and starting to soften but not browned. Why: sweating (not browning) onions gently releases their natural sugars and builds a sweet aromatic base for the soup without introducing bitter roasted flavors.

Add the diced carrots and continue to cook for 6–7 minutes until the carrots soften and begin to take on a little gloss. Why: sautéing the carrots with the onions draws out their sweetness and helps them to finish cooking quickly once the broth is added.

Stir in the garlic and cook for 30–45 seconds until fragrant. Add the dried oregano and basil now and cook for another 20 seconds to 'bloom' them in the fat. Why: garlic burns quickly — add it late and watch closely. Toasting the dried herbs briefly in oil releases their essential oils, deepening the aroma.

Add the chopped tomatoes and tomato paste; stir to combine, then cook for 4–6 minutes until the tomatoes begin to break down and the mixture smells intensely tomatoey. If you want a smoother texture, smash the tomatoes with the back of a spoon as they soften. Why: reducing the tomatoes concentrates their flavor and develops a richer umami base for the soup.

Pour in the vegetable stock, add the sugar (if using) and the kosher salt and pepper. Bring the pot to a gentle simmer, then reduce heat and simmer uncovered for 15–18 minutes so the flavors meld and the carrots finish cooking. Why: simmering (not a rapid boil) allows flavors to marry and prevents the pasta from over-releasing starch if cooked in the same pot later.

While the soup simmers, bring a separate pot of salted water to a boil and cook the pasta to just under the package's al dente time (usually 1–2 minutes less). Why: cooking pasta separately prevents the starchy cooking water from turning the soup gluey and lets you control the pasta texture precisely.

Reserve 1 cup (240 ml) of the pasta cooking water, then drain the pasta. Why: reserved pasta water is a chef's secret — its starch helps loosen and bind the soup to the pasta if you want a more cohesive bowl.

Taste the soup and adjust seasoning. Stir in half the grated Parmesan and a splash (about 1 tsp) of red wine vinegar or lemon juice to brighten. If the soup feels too thick, loosen with a few tablespoons of the reserved pasta water. Why: finishing with cheese adds savory richness; acid at the end wakes up all the flavors and balances the tomato sweetness.

To serve: divide the cooked pasta among 4 bowls, ladle the hot tomato-carrot broth over, then sprinkle with the remaining Parmesan, a crack of black pepper and torn basil leaves. Why: plating the pasta first ensures each bowl gets a perfect proportion of pasta to broth without the pasta soaking and swelling in the pot.

If you prefer, you can ladle soup into the pot with the pasta for a quick toss and serve family-style; add a knob of butter and extra grated cheese at the table. Why: butter mounted at the end gives the broth an emulsion and a luxuriously silky mouthfeel.