Heat oven to 500 degrees F and arrange 1 rack in top and a second rack in middle.
Heat oil in a large frying pan over medium-high heat. When it shimmers, add onions, garlic, and jalapenos, season well with salt and freshly ground black pepper, and cook until softened, about 3 minutes. Add poblanos and stir to coat. Add broth and cook until the liquid reduces enough to just coat the vegetables, about 10 minutes. Remove from heat.
When pepper mixture is cool enough to handle, top 1 tortilla with 1/2 cup cheese and 1/6 of the pepper mixture, close with a second tortilla. Repeat to make 6 quesadillas.
Arrange quesadillas on 2 baking sheets and place 1 sheet on top rack and another on middle rack. Bake until cheese is melted, about 5 minutes. Switch baking sheets and continue to cook until tortillas are golden-brown and crisp, about 5 more minutes.