Beat eggs in bowl with Parmesan cheese, salt, and pepper. Set aside. Dust chicken in flour, then coat in reserved egg mixture.
Melt 1/2 cup butter in a shallow non-stick pan. Add olive oil and cook chicken on both sides over low heat just until browned. If heat is too high, coating may burn and/or stick to pan.
Preheat oven to 350 degrees F.
With a metal spatula, transfer chicken breast halves to a baking dish and bake, uncovered, 15 to 20 minutes or until done.
While chicken bakes, prepare sauce: Combine wine, shallots, and cream. Over high heat, reduce mixture by about half until thickened. Remove from heat.
Cut remaining 1/2 cup butter into small pieces and whisk into the reduction until all the butter is incorporated.
When chicken is done, remove from pan and set on individual serving dishes. Ladle sauce over pieces of chicken, then sprinkle each serving with capers.