Pat the 1 lb chicken breast dry and cut into 1/2-inch-thick strips. If desired, place strips between plastic wrap and lightly pound with a meat tenderizer to even thickness.
Sprinkle salt and pepper evenly over both sides of the chicken strips. Spread the 1 tbsp mustard evenly over both sides of the strips.
Set up two shallow dishes: one with flour for dredging and one with the 2 eggs (beaten).
Dredge each mustard-coated strip lightly in the flour, shaking off any excess.
Dip the floured strip into the beaten eggs, then dredge again in the flour for an even coating. Shake off excess flour.
Pour enough vegetable oil into a skillet to cover the bottom by about 1/8–1/4 inch and heat over medium until the oil shimmers. Place the coated chicken strips in a single layer (do not overcrowd the pan).
Fry the strips 2–3 minutes per side, until golden brown and cooked through. Adjust heat as needed to prevent burning; cook additional batches if necessary.
Transfer cooked strips to paper towels to drain excess oil. Serve immediately.