Trim the top and bottom stump from the head of celery. Rinse well. Set aside the innermost leafy stalks.
Prepare the leeks: Cut off the bottom stump and dark green tops and discard, leaving the white and light green section. Slice the leeks lengthwise through the center. Place the long halves flat-side down on a cutting board, and slice the leeks into 1/4-inch thick semicircles. Rinse well.TIP: You can use the dark green leek tops for chicken stock.
Heat a braising dish or 10" skillet with a lid over medium-low heat with the butter and thyme. When the butter has melted, add the leeks and stir. Stir in 1/4 cup of broth or water, and 1/2 teaspoon of kosher salt. Reduce the heat to the lower side of medium-low and cook, covered, for about 15 minutes, stirring with some frequency, until the leeks are very soft. TIP: The goal here is to not get too much color on the leeks. Check the leeks once or twice while cooking. If they look dry or brown, stir in an additional tablespoon of water.
Remove the lid and give the leeks a stir. Lay the long celery stalks on top of the leeks in a single layer. Add the remaining water or broth and sprinkle with the remaining 1/4 teaspoon of kosher salt and some turns of fresh black pepper.
Cover and cook for about 10 minutes, or until the celery is al dente.
Fish out the thyme sprigs and taste for seasoning. Transfer the celery to a serving platter. If liquid remains in the pan, stir the leeks and cook, uncovered, until most of the liquid has absorbed and the leeks look very melted and saucy.
Spoon the leeks over the celery and garnish the dish with some more black pepper, a squeeze of fresh lemon juice, and some reserved celery leaves. Serve immediately.
Disclaimer: Nutrition information is provided for courtesy purposes only, and is an estimate not verified by medical or nutrition experts. Read the full nutrition disclaimer.