Heat oil in a large pot over medium-high heat. Add ground beef and cook for 6–8 minutes, breaking into pieces, until browned. Spoon off excess fat, leaving about 1 tablespoon.
Add onion and carrot and cook for 5–6 minutes until softened. Stir in garlic, tomato paste, oregano, basil, thyme, and red pepper flakes. Cook for 60–90 seconds until the paste darkens slightly.
Pour in 1 cup of broth and scrape the bottom of the pot. Add remaining broth, crushed tomatoes, ½ teaspoon salt, and black pepper. Bring to a gentle boil, then reduce to medium-low and simmer uncovered for 18–20 minutes.
While the soup simmers, boil lasagna noodles in salted water until al dente (9–12 minutes). Drain and spread loosely in a colander.
Stir spinach into the soup and cook for 1–2 minutes until wilted. Taste and add remaining salt only if needed.
Divide noodles among 6 bowls, ladle the hot soup over them, and top each with a spoonful of the ricotta mixture.