Place the grouse breasts on a cutting board between two sheets of plastic wrap. Using a meat mallet or empty wine bottle, pound the breasts to a thickness of ¼ inch. Don’t go any thinner. Salt and pepper the pounded breasts, dredge them in flour, and shake off excess.
In a cast iron pan, heat up the olive oil and two tablespoons of butter over medium high heat. Cook the breasts on one side until browned. Depending on the size of the pan, you may have to do them in two batches. Flip them over and brown the other side. When done, remove them from the pan and set aside.
Add white wine and lemon juice and turn the heat up to high, scraping up the brown bits with a wooden spoon. Add the capers and then reduce the sauce down for a few minutes.
Turn off the heat. Add the remaining butter to the pan and swirl it around until it's melted and emulsified. It should look smooth and silky.
Plate the grouse breasts. Pour the sauce over them. Garnish with chopped parsley and serve.
