Place chicken breasts between 2 sheets of heavy plastic on a solid, level surface. Firmly pound with the smooth side of a meat mallet to 1/2-inch thickness.
Combine flour, crackers, 1 1/2 tablespoons Parmesan cheese, garlic powder, salt, and pepper in a resealable plastic bag.
Beat eggs in a shallow bowl. Dip chicken breasts into eggs to coat; place in flour mixture and seal bag. Toss chicken in the flour mixture until well covered.
Heat oil in a skillet over high heat. Add chicken breasts; cook until underside is browned and crispy, 4 to 5 minutes. Flip; sprinkle 1 teaspoon Parmesan cheese over each chicken breast. Spoon hot oil over cheese until melted. Drain on paper towels for 2 to 3 minutes before serving.