Rinse the mustard greens thoroughly under cold running water to remove grit. Shake off excess water or spin in a salad spinner. Remove tough stems: either strip leaves from stems and discard or chop stems into 1/2-inch pieces to cook with the onions.

If leaves are large, stack and roll them, then slice crosswise into 1- to 1.5-inch ribbons for even cooking.

Cooking

Sweat aromatics

Heat a large wide skillet or sauté pan over medium heat. Add the olive oil and butter. When the butter melts and the oil shimmers, add the sliced onion (and chopped stems, if using). Cook, stirring occasionally, until the onion is soft and translucent, about 6–8 minutes. Add the minced garlic and crushed red pepper flakes and cook 30–45 seconds more until fragrant, taking care not to brown the garlic.

Add greens & deglaze

Pile the prepared mustard greens into the pan in batches, allowing each batch to wilt slightly before adding more so they fit. Pour in the broth and increase heat to medium-high to bring it to a gentle simmer. Use tongs to toss and submerge the greens so they begin to wilt and incorporate with the aromatics.

Simmer to tender

Reduce heat to medium-low, partially cover the pan, and simmer for 8–12 minutes, stirring occasionally, until stems (if included) are tender and leaves are soft but still bright green. If the pan becomes dry before the greens are tender, add up to 1/4 cup more broth or a splash of water.

Finish & season

Stir in the apple cider vinegar and lemon juice. Season with kosher salt and freshly ground black pepper, taste, and adjust seasoning as needed. If you prefer a milder flavor, add a little more butter or a teaspoon of sugar to balance bitterness.

Serve

Transfer the mustard greens to a warmed serving bowl, spoon up any remaining pan liquid and pour over the top. Sprinkle with chopped toasted nuts and grated Parmesan if using. Serve hot as a side dish.