Add the french fries to a baking sheet and bake until crispy and golden brown, about 12 minutes.
Slice each chicken breast into 2–3 thin filets and season with salt and pepper.
Pound the chicken breast filets into thinner cutlets using a mallet or rolling pin.
Dredge each cutlet in flour, reserving 2 tablespoons of the remaining flour.
Heat the vegetable oil in a large frying pan over medium heat.
Add the cutlets to the hot oil in one even layer (this will be done in 2 batches) and fry until golden brown, about 3 minutes.
Flip and brown on the other side. Remove the cutlets and set aside.
Add butter, garlic, and reserved flour to the skillet and reduce the heat to medium-low.
Add the chicken broth, heavy cream, and Dijon mustard to the butter and stir until incorporated.
Slowly add in the shredded cheese, whisking after each addition, until a thick sauce forms. Set aside.
Place a tortilla on a flat surface and top with ¼ of the french fries, 2 chicken cutlets, 4 slices of ham, and 2 tablespoons cheese sauce. Repeat with the remaining tortillas.
Roll the tortillas into a burrito shape.
Heat a grill pan or panini press on medium-high heat.
Add the tortillas to the pan, press down, and cook for 2–3 minutes.
Flip the tortillas and cook for another 2–3 minutes.
Remove from the grill pan and serve with mayonnaise and ketchup, if desired.