Preheat the oven to 425 degrees F (220 degrees C) and line a sheet pan with foil or parchment. If needed, dry carrots with paper towels or a clean dish towel.
Toss carrots with olive oil and sweet ginger and garlic seasoning. Spread carrots in a single layer on the prepared pan.
Bake in the preheated oven until carrots are just tender, about 20 minutes.
Bring pie crust to room temperature.
Meanwhile, for sauce, melt butter in a 12-inch skillet over medium heat. Stir in applesauce, chutney, and balsamic vinegar. Bring to a boil, then reduce heat and simmer until mixture is thick and coats the bottom of the pan, stirring occasionally, 5 to 8 minutes.
Shake the skillet a bit, to spread applesauce mixture evenly. Arrange carrots in a single layer in a pleasing design, cutting and fitting the carrots as you go, on top.
Remove wrapper of the pie crust, gently unroll, and place on top of carrots. Use a spatula to tuck in the edges of the pastry, where necessary.
Bake at 425 degrees F (220 degrees C) for about 20 minutes, then reduce heat to 375 degrees F (190 degrees C), and continue baking until crust is deeply browned, 15 to 20 minutes.
Select a serving plate that will fit inside the skillet and place upside down on the crust. Holding the plate in place with one hand, use the other hand to flip the skillet, so the tarte Tatin will be crust side down on the serving plate. If any carrots stick to the pan, replace them, prior to serving.
Stir Greek yogurt and honey together for garnish. Serve warm or at room temperature.
