Prepare the ingredients and water Bring approximately 3 quarts water to a boil in the pasta pot. Grate the carrot, dice the onion, mince the garlic, and drain the beans. Put a heatproof measuring cup within reach for reserving pasta water.

Soften the onion Heat the olive oil in the wide skillet over medium heat. Add the onion and ⅛ teaspoon salt. Cook for 4–5 minutes, stirring occasionally, until translucent and softened without dark brown edges.

Cook the pasta and carrot alongside each other Add 1 teaspoon salt and the pasta to the boiling water and cook to al dente according to the package. Meanwhile, add the grated carrot to the onion and cook for 4–6 minutes, stirring, until the strands soften and bend easily. If dry or still firm, add 2 tablespoons plain water, cover for 2 minutes, then uncover.

Add garlic, thyme, and beans Near the end of the pasta cooking, reserve 1 cup cooking water, then drain the ready pasta without rinsing. Reduce the skillet to medium-low. Stir garlic and thyme into the softened vegetables for about 30 seconds. Fold in the beans, black pepper, and ⅓ cup reserved water; warm gently for 2–3 minutes.

Bring the coating together Add the drained pasta and fold gently for 1–2 minutes over low heat. If dry, add reserved water 1–2 tablespoons at a time. Stop when the pasta glistens, the beans are hot, and the carrot is distributed through the pan without a pool of liquid underneath.

Finish with lemon Remove from the heat and stir in 1 teaspoon lemon juice. Taste a bite with pasta, carrot, and beans, then add salt, pepper, or a few more drops of lemon if needed. Divide between two bowls and serve promptly.