Melt a portion of the butter in a 12-inch nonstick skillet over medium-high heat. Once the butter is foaming, sprinkle the sugar and salt evenly across the pan and place the leeks cut-side down in a single layer. Cook until the bottoms are deeply golden, about 5 minutes, reducing the heat if the leeks appear to be browning too quickly.

Pour in the wine and broth, then nestle the thyme sprigs among the leeks. Reduce the heat to low, cover the skillet, and simmer until the leeks are translucent and a paring knife slides into the root end with very little resistance, about 10 minutes.

Transfer the leeks to a serving platter, leaving the liquid behind in the pan. Simmer the remaining liquid over medium-high heat until it reduces to a thick, syrupy consistency, about 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 warm glaze over the leeks and serve immediately.