Preheat the oven to 400°F.
Butterfly the chicken breast by cutting it in half lengthwise to create thin even pieces; place a piece of plastic wrap over the chicken then lightly pound the chicken using a mallet or rolling pin to make the chicken an even thickness and enlarged so that it can be stuffed and rolled; then season both sides with salt and pepper.
Heat the butter in a nonstick pan over medium heat. Add the mushrooms and thyme and saute for 5 - 7 minutes until soft. Remove from heat.
Lay the fontina on the chicken and spoon the mushrooms into the center. Fold in the ends of the breast and roll the chicken up.
Coat the chicken in flour, then dip into the eggs, then roll in the breadcrumbs. Make sure they are coated well.
Heat the oil in a nonstick pan on medium heat. Sear the chicken rolls seam side down first and on all sides until golden brown.
Place the chicken onto a baking sheet seam side down. Bake until an internal temperature of 160 degrees is reached using a thermometer. Let sit for 3 - 5 minutes before slicing.