Beat eggs in a shallow bowl. Combine next four ingredients in another bowl. Turn pieces of fish in crumbs, then in eggs and again in crumbs, patting on crumbs.
Heat 1 Tbsp. each oil and butter in a 12-inch nonstick skillet over medium-high heat. When butter is melted, add 2 pieces of fish and cook, turning once, until golden, about 6 minutes. Add remaining oil and butter to pan and repeat with remaining fish.