Coarsely crush the peppercorns in a spice grinder or with the back of a heavy skillet. Set aside.
Heat the oil and 3 tablespoons of butter in a braiser or large skillet over medium-low heat. When the butter is melted and foamy, add half of the crushed peppercorns and cook until sizzling and fragrant, about 30 seconds.
Add the cabbage and toss to coat with the fat. Season with 1 teaspoon salt. Cover and cook, stirring occasionally, until the cabbage begins to wilt and caramelize, about 8 minutes.
Add 2 cups water to the pan, cover and cook, stirring occasionally, until the cabbage is tender, 10 to 12 minutes.
Uncover and adjust the heat so the sauce is at a lively simmer. If the sauce is still watery, simmer to reduce until it’s saucy.
Stir in the remaining 1 tablespoon butter and the remaining crushed black pepper.
Remove the skillet from the heat and sprinkle with the Pecorino, Parmesan and chives and stir to combine.
Taste and season with salt, if needed. Sprinkle with more Pecorino and Parmesan and serve.
