I love gratin potatoes....this one is a bit of an effort. I think nothing of making the sauce....my grandma was always making these sauces in an attempt to get us to eat veggies. LOL. The combination of sweet and baking potatoes adds a depth of flavor.
Place potatoes in a large stockpot; cover with stock. Bring the mixture to a boil. Cook 4 minutes. Remove from heat. Carefully remove the potatoes from the pot using a slotted spoon, reserving the cooking liquid. Arrange the potato slices in a single layer on a jelly-roll pan; set aside.
Strain the cooking liquid through a fine mesh sieve over a bowl; reserve 1 cup cooking liquid. Discard the solids and the remaining cooking liquid.
Heat a medium saucepan over medium heat. Add the oil to the pan. Sprinkle the flour over the oil; cook 1 minute, stirring constantly with a whisk. Add the garlic; cook 2 minutes, stirring frequently. Combine milk and reserved 1 cup cooking liquid. Gradually pour milk mixture into the flour mixture in pan, stirring constantly with a whisk. Add thyme sprigs to the pan. Bring mixture to a boil; cook 4 minutes or until slightly thick, stirring frequently. Remove from the heat. Strain mixture through a fine-mesh sieve over a bowl, reserving sauce; discard solids. Stir 1/2 teaspoon salt, pepper, and Gruyere cheese into the sauce.
Spread 1/2 cup sauce in the bottom of a broiler-safe 2 quart ceramic baking dish coated with cooking spray. Arrange a single, flat layer of sweet potato and then baking potato slices over the sauce. Over the flat layer, alternate baking potato and sweet potato slices, in shingle-like fashion. Sprinkle evenly with remaining 1/4 teaspoon salt, 1 tablespoon chives, and shopped thyme; pour remaining sauce over the potato mixture. Sprinkle with Parmigiano-Reggiano cheese. Bake at 350 degrees for 1 hour or until the potatoes are tender when pierced with a knife.
Remove gratin from the oven. Preheat the broiler to high.
Place the gratin in the oven. Broil the gratin 3 minutes or until lightly browned. Sprinkle with the remaining 1 tablespoon chives.