Cut cabbage into wedges that are 1 1/2- to 2-inches wide. For a medium head, that's about 8 wedges. Leave the stem intact so that 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 the oil is hot and shimmering, add half the wedges so one of their wide, cut sides is touching the bottom of the pot. Brown on both sides until lightly charred, about 4 minutes per side (I find it easiest to flip the cabbage with a set of tongs). Remove to a plate. Add the remaining 1 1/2 tablespoons oil, then repeat with the remaining wedges, transferring the second batch to the same plate.

Reduce the 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, then increase the heat to bring the liquid to a boil.

Snuggle the cabbage wedges into the pot. Cover, reduce the heat back to medium and cook for 15 minutes, flipping the cabbage once halfway through. The cabbage should be melting and tender. Squeeze the lemon juice over the top, then sprinkle with 1/4 cup of the Parmesan. Turn the cabbage a few times so the Parmesan mixes with the broth. Sprinkle on the remaining 1/4 cup Parmesan, walnuts, and parsley. Serve hot, with as much or as little of the broth spooned over the top as you like.