Melt the butter in a 12-inch nonstick skillet over medium-high heat until the foaming subsides. Whisk in the sugar, salt, and pepper until the sugar dissolves and the mixture is well-combined.

Place the endives in the skillet cut-side down. Cook without moving them until the cut sides are deeply browned, about 3 to 5 minutes.

Pour in the wine and broth, then sprinkle the thyme over the vegetables. Reduce the heat to low, cover the skillet tightly, and simmer until the endives are completely tender when pierced with a paring knife, about 10 to 15 minutes.

Remove the lid and increase the heat to medium-high. Cook until the liquid has evaporated and the remaining juices have thickened into a light golden glaze, about 2 to 4 minutes—watch closely so the glaze doesn't burn.

Remove the skillet from the heat and gently stir in the lemon juice and parsley. Use a spatula and tongs to turn the endives carefully in the glaze to coat them evenly. Serve warm.