In a large skillet, melt 3 tablespoons butter over medium-high heat. Add the mushrooms and cook until softened, 8 to 10 minutes; season with salt and pepper. Transfer to a medium bowl; let cool slightly. Add the ground chicken, 3/4 teaspoon salt and 1/4 teaspoon pepper; shape into 4 patties.
In a large skillet, cook the bacon over medium heat until crisp. Drain, reserving 2 tablespoons of bacon fat in the skillet. Add the onions and cook over medium heat, stirring occasionally, until softened and browned, about 10 minutes.
In a small saucepan, bring the chicken broth, red wine and tomato paste to a boil and cook until reduced to about 1/2 cup, about 8 minutes.
Preheat the broiler. In a large skillet, melt 1 tablespoon butter over medium heat. Add the patties and cook, turning once, for 12 minutes. Meanwhile, spread the remaining 2 tablespoons butter on the rolls and broil until golden. Place a patty on each roll bottom; top with onions, wine sauce, bacon and a roll top.