Season chicken on all sides with salt and pepper. Place milk in a shallow bowl. Combine cornmeal, oregano, 1/4 tsp. salt and cayenne in a separate, shallow bowl.
Melt butter with oil in a large skillet over medium-high heat. Dip chicken in milk, then dredge in cornmeal mixture. Transfer to skillet and pan-fry until browned and cooked through, 3 to 5 minutes per side.
Mix ketchup, Tabasco, sugar and lemon juice in a small bowl. Season with salt and pepper. (Make it up to a week ahead; cover and chill.) Serve chicken warm with dipping sauce.