Season chicken breast with salt and pepper to taste. Heat a 10-inch skillet over medium heat; add 1 tablespoon oil or lard. Cook chicken until browned on both sides and firm when gently pressed with fingertip, about 10 minutes total. Cool slightly; tear chicken into bite-size pieces.
Melt 1 tablespoon butter in skillet over medium heat. Add mushrooms, 1/2 teaspoon salt, cumin, and black pepper, to taste. Cook, stirring occasionally, until mushroom's moisture releases and evaporates, about 4 minutes. Transfer to a small bowl.
Add remaining 3 tablespoons oil or lard to skillet. Add onion, garlic, jalapeno and jar juices, and remaining 1/2 teaspoon salt. Cook until onion is translucent, about 5 minutes. Add beans and mash until almost smooth. Set aside.
Place tortillas on work surface and spread 1/4 of bean mixture on one side of each tortilla (reserve leftover bean mixture for later use). Top with 1/4 of cheese, half of chicken, and half of mushrooms. Sprinkle with remaining cheese. Fold tortillas over to encase filling.
Heat remaining 1 tablespoon butter in a skillet, fry each quesadilla until golden on both sides, about 4 minutes. Cut into wedges with a pizza wheel or serrated knife. Serve with guacamole and salsa.
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