Preheat oven to 350 degrees and line a large baking sheet with parchment paper. Place a chicken breast between two pieces of plastic wrap on a cutting board and flatten to a 1/4" thickness with a meat mallet or rolling pin. Top chicken with 1 slice of cheese, then 1 slice of ham. Starting at the top of the breast, roll up tightly and secure with toothpicks. Repeat with remaining chicken breasts.

Place flour, eggs, and bread crumbs in three shallow bowls. Season flour with salt and pepper. Add melted butter and oregano to breadcrumbs and mix until incorporated.

Working with one at a time, roll chicken first in flour, then eggs, then breadcrumbs mixture, pressing to coat. Place on prepared baking sheet. Bake until golden and cooked through, 30 minutes.

For the sauce: In a large skillet over medium heat, melt butter. Add flour and stir until golden, 2 minutes. Slowly whisk in milk, then add mustard. Let simmer until thickened, 5 minutes. Add Parmesan and stir until melted. Season with salt and pepper.

Remove toothpicks from chicken and serve with sauce.