Step 1

In a large Dutch oven, heat 4 tablespoons oil over medium-high. Working in batches if necessary, add the cabbage wedges cut side down in a single layer, season with salt and cook until well browned, 3 to 5 minutes per side. Transfer to a cutting board to cool slightly. Repeat with remaining cabbage, adding more oil as needed.

Step 2

Reduce the heat to medium-low and add the remaining 2 tablespoons oil and the onion to the empty pot. Season with salt and pepper, and cook, stirring occasionally, until softened and starting to turn golden, 5 to 7 minutes. Bring a large pot of salted water to a boil. Coarsely chop the cooked cabbage, including the core.

Step 3

Reserve 1 teaspoon of the garlic, and add the remaining garlic and the tomato paste to the onions. Cook, stirring constantly, until the tomato paste is a shade darker, 3 to 5 minutes. Add the wine and cook, scraping up browned bits, until nearly evaporated, 1 to 2 minutes.

Step 4

Add the chopped cabbage, 1/4 cup parsley and the sugar, and stir to coat in the tomato paste. Season with salt. Add 1/4 cup water, cover and cook, stirring occasionally, until the cabbage is soft and shiny, 15 to 20 minutes. (If the cabbage is sticking to the pot, add 1 tablespoon water and reduce the heat.)

Step 5

In a small bowl, stir the reserved 1 teaspoon garlic into the remaining 1/2 cup parsley. Add the red-pepper flakes, if using.

Step 6

When the cabbage is just about done, add the pasta to the boiling water and cook until al dente. Reserve 2 cups pasta water, then drain the pasta. (If the cabbage is ready before the pasta, keep covered off the heat.) Add the pasta and 1 cup pasta water to the cabbage. Stir over medium-low until the pasta is glossed with sauce. Add more pasta water as needed to thin the sauce. Divide among bowls and top with parsley-garlic mixture.