Place each chicken breast between plastic wrap and pound to a uniform 1/4-inch (6mm) thickness using a meat mallet.
Combine salt, pepper, and garlic powder, then sprinkle evenly over each pounded chicken breast.
Lay 2 slices of ham and 2 slices of Swiss cheese on each seasoned breast, leaving a 1/2-inch border along the far edge for sealing.
Roll the chicken tightly from the short end, tucking in the sides, and secure with 2-3 toothpicks inserted at an angle. Refrigerate for 30 minutes if desired.
Set up a breading station with three bowls: seasoned flour, whisked eggs with milk, and panko with salt and pepper.
Coat each roll first in flour, then egg wash, and finally press firmly into panko crumbs for an even coating.
Heat oil in a large oven-safe skillet over medium-high heat. Sear the breaded rolls seam-side down for 2 minutes per side until golden brown.
Preheat oven to 375°F (190°C), then transfer skillet to oven and bake for 18-22 minutes until internal temperature reaches 165°F (74°C).
While chicken bakes, melt butter in a small saucepan and sauté shallot until softened. Add garlic and flour, cook 1 minute, then whisk in broth, cream, Dijon mustard, and thyme. Simmer 4-5 minutes until thickened.
Rest chicken for 5 minutes, remove toothpicks, and serve with the warm cream sauce spooned over the top.