Melt the butter in a 12-inch nonstick skillet over medium-high heat until the foam subsides. Sprinkle the sugar and salt evenly into the skillet, then lay the endives cut-side down in a single layer. Cook for about 5 minutes, shaking the skillet occasionally, until the first side is golden brown; adjust the heat if the endives brown too quickly.

Turn the endives over and cook until the curved sides are lightly golden, about 3 minutes.

Flip the endives cut-side down again and add the wine, broth, and thyme. Reduce the heat to low, cover, and simmer for 13 to 15 minutes, checking occasionally, until the leaves open up slightly and the endives are tender throughout when poked with a paring knife. If the pan appears dry during the simmering process, add a splash of water.

Transfer the endives to a platter and increase the heat to medium. Simmer the liquid remaining in the skillet until it reaches a thick, syrupy consistency, about 1 to 2 minutes.

Remove the skillet from the heat and whisk in the remaining butter, parsley, and lemon juice. Season with salt and pepper to taste, then spoon the sauce over the endives and serve warm.