Heat oil in a skillet over medium-high heat. Cook mushrooms and onion for 7 to 8 minutes until their released water evaporates and the pieces start to brown.
Add turkey and cook for 5 to 6 minutes, breaking it into small crumbles.
Stir in tomato sauce, ketchup, vinegar, Worcestershire, brown sugar, mustard, salt and pepper. Simmer for 10 to 12 minutes until thick and the turkey reaches 165°F.
Toast the cut sides of the buns in a dry skillet for 1 to 2 minutes.
Divide the hot turkey filling among the buns and serve immediately.