Place potatoes in a large stockpot; cover with stock. Bring mixture to a boil; cook 4 minutes. Remove from heat. Carefully remove potatoes from pot using a slotted spoon, reserving cooking liquid. Arrange potato slices in a single layer on a jelly-roll pan; set aside. Strain cooking liquid through a fine-mesh sieve over a bowl; reserve 1 cup cooking liquid. Discard solids and remaining cooking liquid.
Heat a medium saucepan over medium heat. Add oil to pan. Sprinkle flour over oil; cook 1 minute, stirring constantly with a whisk. Add garlic; cook 2 minutes, stirring frequently. Combine milk and reserved 1 cup cooking liquid. Gradually pour milk mixture into flour mixture in pan, stirring constantly with a whisk. Add thyme sprigs to pan. Bring mixture to a boil; cook 4 minutes or until slightly thick, stirring frequently. Remove from heat. Strain mixture through a fine-mesh sieve over a bowl, reserving sauce; discard solids. Stir 1/2 teaspoon salt, pepper, and Gruyère cheese into sauce.
Spread 1/2 cup sauce in 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 sauce. Over 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 chopped thyme; pour remaining sauce over potato mixture. Sprinkle with Parmigiano-Reggiano cheese. Bake at 350° for 1 hour or until potatoes are tender when pierced with a knife.
Remove gratin from oven. Preheat broiler to high.
Place gratin in oven. Broil gratin 3 minutes or until browned. Sprinkle with remaining 1 tablespoon chives.