Pulse the mushrooms in a food processor until finely chopped.
Heat the olive oil in a large skillet over medium-high heat until shimmering. Add the onion, bell pepper and 1/2 teaspoon each salt and pepper and cook, stirring occasionally, until slightly softened, about 4 minutes. Add the mushrooms and an additional 1/2 teaspoon salt and cook for 3 minutes. Add the ground turkey, 1 teaspoon salt and some pepper and cook, stirring and breaking up any clumps with a wooden spoon, until the mushrooms have released their liquid and the meat is browned, about 8 minutes. Add the crushed tomatoes, chili powder, coconut aminos, Worcestershire and apple cider vinegar. Bring to a simmer and cook, stirring occasionally, until slightly thickened, about 5 minutes. Serve on buns.