Set up an assembly line of three rimmed dishes. In the first dish, add the flour, salt, and pepper; mix well. In the second dish, add the 2 beaten eggs. In the last dish, combine the panko and parmesan cheese.
Preheat oven to 350° F. Line a large jelly roll pan/baking sheet with foil, and top with an oven-safe cookie cooling rack. Spray well with non-stick cooking spray. Set aside.
Heat 1 tablespoon each of the butter and olive oil in a non-stick skillet over medium-high heat. Once the pan is nice and hot, pan sear the chicken (only 2 at a time to avoid over-crowding) for 2-3 minutes per side. Keep a close eye as they will brown quickly due to the parmesan cheese. They should be nice and golden when you flip them. Once both sides are browned, remove and place onto prepared pan. Add the remaining tablespoon of butter and olive oil and repeat process with the last 2 chicken breasts.
Place pan into oven and bake for 12-18 minutes at 350º F until chicken is cooked through (cooking time depends on how thin you pound out the chicken).
For the Lemon Butter Sauce:
While chicken is baking, add the butter, olive oil, and chicken broth to the pan that the chicken was cooked in. Bring to a simmer and whisk the flour into it until all of the lumps have been removed and it begins to thicken. Season with oregano, salt, and pepper. reduce heat to low. Once chicken is about ready, add the half and half and lemon juice to the sauce and whisk together until warmed through. Serve chicken over mashed potatoes with a drizzle of the sauce. Delicious!