In a medium bowl, stir together turkey, basil, salt, pepper, and garlic powder. Form into 4 patties.
In a large nonstick skillet, heat 1 T. oil over medium-high heat, swirling to coat the bottom. Cook the patties for 5 minutes; turn over and cook for 4 minutes, or until no longer pink in the center. Transfer to a plate, and cover to keep warm.
Still on medium high, heat the remaining 1 1/2 teaspoons oil in the skillet, swirling to coat the bottom. Cook the mushrooms, onion, and cayenne for 4 minutes, or until the onion is soft, stirring frequently. Remove from heat.
To serve, spread the mustard over the patties, and top with the mushroom mixture.