Peel and trim the red onions. Slice each onion in half through the root, then place cut-side down and slice into 1/8- to 1/4-inch (3–6 mm) half-moon strips so you have even slices for uniform cooking.

Heat pan

Place a large skillet over medium heat. Add the olive oil and butter and allow the butter to melt and foam but not brown, about 1 minute.

Add onions

Add the sliced red onions to the skillet. Stir to coat with oil and butter so the slices form an even layer. Spread them out so they contact the pan.

Sweat and soften

Reduce heat slightly to medium-low. Cook, stirring every 2–3 minutes, until the onions are softened and translucent, about 8–10 minutes. If they begin to brown too quickly, lower the heat.

Caramelize

Sprinkle the sugar evenly over the onions to encourage caramelization. Continue cooking, stirring occasionally and scraping any brown bits from the pan, until the onions turn a deep rosy-brown and develop sweet flavor, about 10–15 more minutes. If the pan becomes dry or the onions start to stick, add 1–2 tablespoons of water or stock to deglaze.

Season and deglaze

Season with sea salt and freshly ground black pepper. Pour in the red wine vinegar and the remaining 1–2 tablespoons of water or stock. Stir and cook 1–2 minutes more to let the vinegar reduce and coat the onions, tasting and adjusting seasoning as needed.

Finish

Remove from heat. If using, stir in chopped fresh thyme or parsley. Allow the onions to cool slightly before serving; they will keep their glossy texture and concentrated flavor.

Optional finishing

To serve, scatter crumbled feta or goat cheese over the warm caramelized red onions for a salty-creamy contrast, or use the caramelized onions as a topping for burgers, sandwiches, salads, grilled meats, or pizzas.