Soft Scrambled Eggs with Creamy Leeks

Even if you reduce the number of eggs to make this dish for fewer people, make the full batch of leeks.

In a large mixing bowl, whisk together the eggs and 2 tablespoons of the milk until light and evenly mixed. You will feel a resistance when you start whisking the eggs, but when they are fully combined, that resistance will give. Set aside.
Halve the leeks lengthwise and slice into thin half moons. Warm the butter in a large sauté pan over medium heat. Add the leeks and ½ teaspoon of salt and stir to coat. Sauté, stirring occasionally, until the leeks are softened and browned in parts, 12 to 15 minutes. Add the remaining 4 tablespoons of milk and the thyme and cook until the liquid is absorbed into the leeks. Turn off the heat and set aside.
In a large nonstick pan, preferably 12 inches, heat the olive oil over mediumlow heat. Give the eggs one more whisk. When the pan is warm, tilt it so the oil puddles in the corner closest to you and slowly pour in the eggs into the center of the oil, so the oil spreads outward with the eggs as you lower the pan down. Cook the eggs without stirring until the bottom just starts to set, then gently push the cooked eggs from the edges of the pan toward the center, letting the uncooked parts fall back toward the edges. Continue until the eggs are almost cooked, turn off the heat, add a pinch or two of salt and pepper and give them another push to make sure any uncooked parts touch the bottom of the pan. Immediately transfer to plates, garnish with more thyme, and serve with the leeks.