Add the oil to a medium skillet over medium-high heat. When the oil is hot, add the mushrooms and cook, without stirring, until the undersides are browned, 2 to 3 minutes. Stir and let brown on the other side, about 2 minutes, then reduce the heat to medium and add the shallots and garlic. Cook and stir until the shallots are softened, about 5 minutes. Remove from the heat, then cool the mixture.
Combine the turkey, yogurt, breadcrumbs, egg and cooled mushrooms in a large bowl. Season with salt and pepper. Mix together lightly with a wooden spoon just to combine; don't overmix. Form into four 1-inch-thick patties. Refrigerate for 2 to 3 hours and up to overnight.
Preheat a grill or grill pan to medium-high heat. Sprinkle the outsides of the patties with salt and pepper. Brush the grill grates with oil and grill the burgers until nicely marked, cooked through and the internal temperature reads 155 to 160 degrees F, about 6 minutes per side. Serve on the buns with additional yogurt, tomato and cucumber slices.