Use a sharp knife to trim off the leek greens right where the light green ends and the dark green parts begin. Set aside the dark greens for later use. Slice each leek in half lengthwise. Trim off the hairy end of the leek (if it has it - some don't) but keep the root end intact.
When cleaning leeks, it's important to get between the layers because that's where the dirt gathers. So thoroughly wash each half under cold running water to remove the dirt and grit. Shake off any excess water, pat dry with a clean dish towel or paper towel, and set aside.
Heat a large skillet (that has a lid you'll use later) over medium heat. Once warm, add a drizzle of olive oil along with the sugar and salt. Give it a quick stir and add the leeks in a single layer with the cut side down. Cook them for 5 minutes on the first side, flip, and cook for 2-3 more minutes.
Add the broth, lemon juice, and fresh thyme sprigs. Cover with a lid and reduce to medium-low heat. Cook covered for 25 minutes undisturbed.
After the 25 minutes are up, remove the lid and the sprigs of thyme, increase to medium-high heat, and boil until the liquid has reduced by half.
To serve, drizzle the reduced liquid in the pan over the top of the leeks and top with flaky sea salt and any additional ingredients such as lemon zest or fresh herbs.