Cheddar sauce:

1. In a saucepan, melt the butter. Stir in the flour and cook, stirring constantly, until combined. Remove from heat. Gradually add milk until the sauce is nice and smooth. Return to heat and, stirring constantly, bring to a boil. Reduce heat and simmer 8-10 minutes. Sprinkle with a pinch of salt and pepper. Add cheese and heat until melted, but do not let the sauce boil.

Rest of sandwich:

1. While cheddar sauce is reducing, start on the rest of the sandwich. If you plan to serve it with the oven fries, start those about an hour before you plan to eat.
2. Slice chicken into 1/2 inch thick strips. Dip in egg and then in flour (or flour, then egg then flour again). Put some olive oil in a skillet and cook over medium-high heat 4-6 minutes per side until no longer pink in the center, and golden brown on the outside.
3. Thinly slice the tomato and red onion.
4. Warm the rolls in the oven for a few minutes if desired.
5. Layer the sandwich: roll, chicken, lettuce, tomato, cheddar sauce, then lettuce.
6. Serve immediately with oven fries if desired.