Heat the olive oil in a large pan over medium heat. Add the carrot and onion, and cook until the vegetables have softened slightly, 8-10 minutes.
Add the beef and season with the oregano, red pepper flakes, and pepper. Use a wooden spoon to break the beef into smaller pieces, and cook until browned, about 10 minutes.
Add the garlic and cook until fragrant, 2-3 minutes.
Deglaze the pan with red wine, scraping up any browned bits from the bottom of pan.
Once the wine has reduced by half, stir in the crushed tomatoes. Cook until sauce has thickened, stirring frequently, about 10 minutes.
Stir in the basil, then remove the sauce from the heat and let cool.
Bring a large pot of salted water to a boil. Working in batches, add the cabbage leaves, using tongs to submerge them in the water. Cook until the leaves have softened and are flexible, 1-2 minutes. Drain well and pat dry with a paper towels.
Stir the egg into the cooled meat sauce.
Place the largest cabbage leaf on the bottom of a 9-inch (23-cm) round baking dish. Line the exposed sides and bottom of the pan with more cabbage leaves.
Spoon in ¼ of the meat sauce, spreading evenly. Dollop ¼ of the ricotta on top of the sauce. Cover with cabbage leaves. Repeat with the remaining sauce, ricotta, and cabbage to make 3 more layers.
Fold any overhanging cabbage leaves toward the center of the pan.
Bake on the middle rack of the oven for 1 hour, until lightly browned on top. Let cool for 20 minutes.
To serve, place a serving plate over the baking dish and invert the chou farci onto the plate. Remove the pan, then slice into wedges and serve.