Heat oil in a large nonstick skillet over medium-high heat. Add onion and cook, stirring often, until soft and beginning to brown, 2 to 3 minutes. Add mushrooms, bell pepper, oregano and pepper and cook, stirring often, until the vegetables are wilted and soft, about 7 minutes.
Reduce heat to low; sprinkle the vegetables with flour and stir to coat. Stir in broth and soy sauce; bring to a simmer. Remove from the heat, lay 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 and serve immediately.