Cut cabbage into wedges that are 1 1/2 to 2 inches wide, leaving the stem intact so the wedges hold together.

Heat a Dutch oven or similar deep, heavy bottom pot over medium high heat. Add 1 1/2 tablespoons of the oil. Once hot, add half the wedges with one of their wide, cut sides touching the bottom. Brown on both sides until lightly charred, about 4 minutes per side. Remove to a plate. Repeat with the remaining wedges.

Reduce heat to medium. Add the butter and onion. Cook, stirring occasionally, until the onion softens, about 4 minutes.

Add the garlic and salt. Cook, stirring constantly, until the garlic is fragrant, about 30 seconds.

Pour in the broth and increase the heat to bring the liquid to a boil.

Snuggle the cabbage wedges into the pot. Cover, reduce heat back to medium, and cook for 15 minutes, flipping the cabbage once halfway through.

Squeeze the lemon juice over the top and sprinkle with 1/4 cup of the Parmesan. Turn the cabbage a few times to mix the Parmesan with the broth. Sprinkle on the remaining 1/4 cup Parmesan, walnuts, and parsley. Serve hot with broth spooned over the top.