Heat 2 teaspoons oil in a large nonstick skillet over medium-high heat. Add sliced onion; cook, stirring often, until soft and beginning to brown, 2 to 3 minutes.
Add sliced mushrooms, sliced bell pepper, 2 tablespoons fresh oregano (or 2 teaspoons dried) and ½ teaspoon pepper; cook, stirring often, until the vegetables are wilted and tender, about 7 minutes.
Reduce heat to low; sprinkle the vegetables with 1 tablespoon flour and stir to coat.
Stir in ¼ cup broth and 1 tablespoon soy sauce; bring to a simmer. Remove from heat.
Lay 3 ounces cheese slices on top of the vegetables; cover and let stand until melted, 1 to 2 minutes.
Divide the mixture into 4 portions with a spatula, leaving the melted cheese layer on top. Scoop a portion onto each toasted bun.
