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In a small bowl, whisk together eggs and cheese.
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Heat 1 tbsp. of oil in a large skillet over medium heat. Dip 3 chicken breasts in egg mixture until well coated on both sides. Cook for 3 to 5 minutes on each side, or until internal temperature reaches 165 degrees. Remove from skillet and keep warm. Heat another 1 Tbsp. of oil in the same skillet. Repeat with the remaining 3 chicken breasts.
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Meanwhile, brush remaining 1 tbsp. of oil on both sides of bread slices. In a separate large skillet over medium heat, toast bread slices for about 2 minutes on each side.
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Meanwhile, in a small saucepan over medium heat, cook gravy for 3 to 5 minutes, or until heated through, stirring occasionally.
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To serve, place a chicken breast on each toasted bread slice. Spoon warm gravy over top and serve immediately. Garnish with fresh thyme sprigs if desired.