Pre-heat oven to 375 degrees.
Remove root from leeks. Slice into thin rounds until just before the dark green part of the leek. Add rounds to a large bowl, pushing rounds out to separate as you add them to a bowl. Fill the bowl with cold water and let leeks soak for 10 minutes.
Meanwhile, chop the small potatoes into quarters (about 1 inch) or evenly sized pieces. Chop the onion and set aside.
Prep the vinaigrette by adding olive oil, lemon juice, 1 3/4 teaspoons of kosher salt, 2 teaspoons of thyme, and ground mustard to a small bowl of small pyrek with a pour spout. Whisk vigerously.
Season both sides of the chicken breasts with 1 teaspoon of kosher salt, 1/4 teaspoon of black pepper, and 1 teaspoon of thyme. Place in baking dish.
Drain leeks. Return to bowl and add potatoes, onions, and peeled garlic cloves. Mix.
Give the vinaigrette another whisk and then drizzle over the bowl and mix well so that all of the leeks and potatoes are coated. Be sure that it all is poured on in case the salt has settled to the bottom of the dish. Add to dish with chicken, surrounding chicken.
Bake for 45 minutes to 1 hour, until the internal temperatue of the chicken reads at least 165 degrees.