Prep: 35 minutes; Cook: 1 hour, 15 minutes.
Preheat oven to 375°. Line a large square baking dish with foil and place in middle of oven. Spray a 9 1/2- to 10-inch fluted tart pan with cooking spray; set aside.
Dust work surface lightly with flour. Unroll pie crust and roll dough into a 12-inch round. Fit in tart pan and press overhang inward to reinforce side. Prick bottom in several places with a fork. Chill 30 minutes (or freeze 10 minutes).
Place a piece of parchment over crust in tart pan and fill with pie weights. Bake in oven until dough just sets (15-20 minutes), then remove parchment and weights and bake until bottom is pale golden (5-10 minutes more). Remove from oven and immediately brush bottom with egg white; cool completely.
Mince and mash garlic to a paste. Add to a bowl along with cream, milk, nutmeg, salt, and pepper. Whisk to combine.
Layer potatoes, alternating Yukon Golds and sweet potatoes, in tart pan. Pour cream mixture over potatoes, pressing gently to distribute liquid. Sprinkle top with Parmesan and dot with butter. Cover surface directly with a round of parchment paper.
Bake until potatoes are tender (test with a small sharp knife), 40-50 minutes. Remove parchment and bake until browned in spots (15-20 minutes). Cool in pan on rack 20 minutes before removing side of pan.