Combine bread crumbs, paprika, pepper and flour on a paper plate.
Dip chicken in egg, making sure all sides are coated, and then dredge in crumb mixture. Repeat process with second chicken breast, being sure to use all the egg and crumbs. Set aside.
In 10 inch skillet, combine oil and butter; heat over medium heat until bubbly and hot. Add chicken and cook, turning once, until both sides are lightly browned and chicken is almost cooked through, 2 to 3 minutes on each side; transfer to plate and keep warm.
To same skillet, add mushrooms and saute until lightly browned, 1 to 2 minutes; sprinkle with 1 tsp flour and stir quickly to combine. Add chicken broth and wine and cook, stirring constantly, until mixture thickens, about 1 minute.
Return chicken to skillet and top evenly with cheese; cover and cook until cheese is melted, 2 to 3 minutes.