Mix egg and milk together in a shallow bowl. Mix flour, salt, and paprika together in a second shallow bowl. Dip chicken into egg mixture, then lift so excess egg drips back into the bowl. Press into flour mixture to coat both sides.

Melt butter in a large skillet over medium-high heat. Add chicken and sauté until golden brown, 2 to 3 minutes per side. Add mushrooms and onion; sauté for 3 to 5 minutes.

Mix broth, wine, lemon juice, and cornstarch together in a medium bowl; pour over chicken and mushrooms. Reduce heat to medium-low and simmer until chicken is cooked through and juices run clear, about 25 minutes. An instant-read thermometer inserted into the center should read at least 165°F (74°C).