Melt 1 tablespoon butter in a skillet. Cook chicken strips until no longer pink in the center, about 10 minutes. Season with salt and pepper. Transfer chicken to a plate.
Melt the remaining butter in the same skillet over medium-low heat. Whisk in flour until smooth. Cook, whisking constantly, until well combined, about 1 minute. Increase heat to medium. Add milk gradually. Cook, stirring constantly, until thickened and bubbling, about 7 minutes. Add shredded cheese, salt, and pepper. Cook and stir the sauce until the cheese melts, about 2 minutes. Remove from heat.
Beat egg with 1 tablespoon water in a small bowl.
Lay 1 egg roll wrapper on a work surface. Fold 1 piece of ham in half and place in the center of the wrapper. Top with a half-slice of Swiss cheese and a chicken strip. Cover with another half-slice of Swiss cheese and folded piece of ham. Brush the edges of the wrapper with the egg mixture; fold wrapper in half over the filling and seal the edges. Repeat with the remaining wrappers.
Heat oil in a deep-fryer or large saucepan to 350 degrees F (175 degrees C).
Place egg rolls in the hot oil; fry until crispy, about 3 minutes. Drain on paper towels.
If cooking whole breast halves, dice and shred the chicken afterwards and spoon about 1/3 cup of it into each egg roll.
We have determined the nutritional value of oil for frying based on a retention value of 10% after cooking. Amount will vary depending on cooking time and temperature, ingredient density, and specific type of oil used.