Preheat the oven to 425˚. Spread the tots on a baking sheet and bake until crisp, 20 to 25 minutes.
Meanwhile, heat the olive oil in a large nonstick skillet over medium-high heat. Add all but 1/4 cup white onion; cook until softened, about 3 minutes. Stir in the turkey, chili powder, pie spice, a big pinch of salt and a few grinds of pepper. Cook, stirring and breaking up the meat, until the turkey is well browned, 4 to 5 minutes. Add the chili sauce, Worcestershire sauce and 1/2 cup water; stir to combine. Reduce the heat to medium and simmer, stirring frequently, until the turkey is fully cooked and the sauce is thickened slightly, 3 to 5 minutes; season with salt and pepper.
Divide the turkey mixture among the bun bottoms. Sprinkle with the cheese, reserved chopped onion and the bun tops. Serve with the veggie tots.