Heat oil in a large skillet over medium-high heat. Season turkey evenly with 3/4 teaspoon salt and 1/4 teaspoon pepper. Combine remaining 1 1/4 teaspoons salt, 1/4 teaspoon pepper, and flour in a shallow dish, stirring well.Combine milk and egg in a separate shallow dish, stirring well with a whisk.Dredge turkey in flour mixture. Dip in egg mixture; dredge in flour again.
Add breaded turkey to pan; cook 3 minutes on each side or until turkey is done. Remove turkey to a paper towel-lined plate using a slotted spoon. Remove and discard all but 1 1/2 tablespoons of oil and drippings in pan; return pan to medium heat. Add wine to pan; bring to a boil, stirring well. Cook until liquid almost evaporates. Stir in flour; cook 1 minute, stirring constantly.
Combine milk and broth; add to pan. Bring mixture to a boil; cook 2 minutes or until desired consistency. Stir in the salt and pepper. Serve gravy with turkey.