Melt butter in a skillet over medium heat. Add next 4 ingredients and cook until onions are translucent. Transfer to a bowl and allow to cool. Stir in cheese and breadcrumbs. Lay pheasant breasts on a firm surface and lightly pound on the thickest parts until the overall thickness is about even with the thinnest part (read it a couple of times and it might make sense!). Take a very sharp knife and cut a pocket into the side of the pheasant breast. Work your knife into the breast, but be careful not to slice through. Divide the mushroom stuffing into 8 equal portions and stuff each breast evenly. Rub pheasant with 2 tablespoons of the olive oil and season with salt and pepper. Pheasant can be baked, grilled or sautéed, but not overcooked!