Lay the chicken breasts on a cutting board and slice each diagonally into 4 or 5 large strips.
Combine the flour, salt, and pepper on a dinner plate. Beat the eggs with 1 tablespoon of water on a second plate. Combine the breadcrumbs and Parmesan on a third plate. Dredge the chicken breasts on both sides in the flour mixture, then dip both sides into the egg mixture and roll in the bread-crumb mixture, pressing lightly to coat.
Heat 1 tablespoon of butter and 1 tablespoon of olive oil in a large saute pan and cook the chicken strips on medium-low heat for about 3 minutes on each side, until just cooked through. Don't crowd the pan. Add more butter and oil and cook the rest of the chicken breasts. Serve each strip on a skewer or stick.
You can keep the chicken breasts warm for about 15 minutes on a sheet pan in a preheated 200 degree F oven.