Heat 1 tsp oil in skillet over medium; thinly slice onion + sauté 8-10 min until browned; season with salt + add 2 tbsp water; cover + cook 3-5 min until softened

Meanwhile, wipe 4 mushrooms with a damp paper towel, remove stems + scoop out gills; heat 1 tbsp oil in a skillet over med-high; cook 2-3 min per side until browned + tender; add salt/pep + set aside

Form the turkey into 2 thin patties; heat 1 tsp oil in skillet over med-high, add patties, season with salt/pep + cook undisturbed until bottoms brown, about 2 min; flip over + cook 2-4 min until no pinkness remains

Divide burgers to mushroom buns + top with caramelized onions