Melt butter in a skillet over medium-high heat. Cook mushrooms with salt and pepper for 7 to 9 minutes until their liquid evaporates and edges brown.
Add turkey ribbons and broth, cover and warm over low heat for 3 to 5 minutes until the filling reaches 165°F. Uncover briefly to evaporate loose liquid.
Heat the broiler and place muffin halves cut-side up on a broiler-safe sheet. Toast for 1 to 2 minutes until lightly crisp.
Divide hot turkey and mushrooms over bottom muffin halves and add Fontina. Broil for 1 to 2 minutes until cheese melts, removing the plain top halves if they brown sooner.
Close with the toasted muffin tops, rest for 1 minute and serve one melt per person.