Line a baking sheet with parchment paper or spray with nonstick cooking spray. Set aside.
Put each chicken breast between two pieces of plastic wrap that are 4 inches wider than the chicken breast. Evenly pound with the flat side of a meat mallet, being careful not to tear the chicken, until it is 1/4 inch thick. Remove the top piece of plastic wrap and discard. Lay 1 to 2 slices of ham on the chicken breast to cover. Top with one-fourth of the cheese and spinach leaves.
Roll the chicken breast up like you would roll a burrito: Using the bottom piece of plastic wrap as an aid, fold the bottom of the breast up about an inch, then fold in the sides. Roll tightly.
Wrap in the plastic wrap and tightly twist the ends to shape and compress the chicken. Repeat with the remaining chicken breasts. Refrigerate for 1 hour.
In the meantime, season the flour with salt and pepper and put in a shallow dish. Combine the breadcrumbs, Parmesan cheese, butter, and thyme. Season with salt and pepper and put in a second shallow dish. Whisk the eggs in a third dish.
Arrange on your workspace in this order: flour; eggs; breadcrumbs. Put the prepared baking sheet next to the breadcrumbs.
Remove the plastic wrap from the chicken breasts. Coat each lightly with flour then, dip in the egg. Finally, roll in breadcrumbs, patting them to make them adhere. Arrange on the baking sheet.
When ready to cook, start the Traeger grill on Smoke with the lid open until the fire is established (4 to 5 minutes). Set the temperature to 350F and preheat, lid closed, for 10 to 15 minutes.
Place the baking sheet with the chicken on the grill grate. Bake for 25 to 30 minutes, or until the coating is golden brown and the chicken is cooked through.
Serve whole, or slice crosswise into pinwheels with a sharp serrated knife.