How to Make It
Place beaten egg in a shallow bowl or pie plate. Spread flour in a separate shallow bowl or pie plate. Season chicken with salt and pepper. Working with 1 breast at a time, dip chicken in egg, letting any excess drip back into bowl, then dredge in flour, shaking off excess.
Melt 1/4 cup butter with oil in a 10-inch stainless steel skillet over medium-high, and cook, swirling occasionally, until milk solids begin to sink to bottom of skillet and start to brown, 4 to 5 minutes. Add chicken, rounded sides down, and cook, swirling skillet occasionally, until lightly golden, about 3 minutes. Carefully turn chicken over, and add remaining 6 tablespoons butter. Tilt skillet toward you so butter pools, and cook, basting chicken often, until chicken is just cooked through and golden brown, 4 to 5 minutes.
Remove from heat, and carefully pour in lemon juice (butter will start to bubble). Serve immediately.