Melt the butter in a small skillet or saucepan over medium-low heat. Cook, swirling occasionally, until the milk solids turn light brown, about 5 minutes. Add the ginger and stir for a few seconds. Remove the pan from the heat, add the lime zest and juice, and swirl to combine.
Heat the oil in a 12-inch nonstick skillet over medium heat. Add the Brussels sprouts and 1 tsp. salt; toss well to coat with oil.
Cover the pan with the lid ajar by about 1 inch. Turn the heat down to medium low, and cook, stirring occasionally, until the sprouts start to brown, 8 to 12 minutes. As the mixture cooks, you should hear a gentle sizzle.
Uncover, turn the heat down to low, add the leeks (with any water still clinging to them) and cook, stirring occasionally and then pushing the vegetables back into a single layer so that most have direct contact with the pan, until the leeks are limp and the sprouts are well browned, about 15 minutes.
Remove the pan from the heat. Pour the butter mixture over the sprouts and leeks and stir, scraping the bottom of the pan and tossing well to coat. Season to taste with salt. Serve right away or let sit off the heat, partially covered, until ready to serve, and then reheat gently over medium-low heat.