Preheat oven to 350°F. Lightly grease a baking dish and set aside.
Heat oil in a large skillet. Dredge pheasant breasts in seasoned flour, then dip in beaten egg mixture. Place breasts in hot oil and sauté until brown for about 3 to 4 minutes each side, turning once. Place breasts in oiled baking dish and place in the preheated oven. Lay a piece of foil on top of baking dish but do not crimp or cover tightly. Bake for about 12 to 15 minutes.
While pheasant is baking, sauté onion in the skillet for 3 or 4 minutes. Carefully add vermouth and turn heat to high and reduce by half. Add olives and butter. Place baked pheasant breasts in the skillet and cook over medium heat for about 5 minutes or until meat thermometer registers about 160°F to 165°F. Spoon sauce over breasts while cooking. Skillet may be lightly covered.
Serve warm. May also be made ahead and refrigerated overnight and reheated. If doing so, do not overcook the pheasant on the first go-round.