Preheat the oven to 400ºF. Cut 8 pieces of kitchen twine into 6-inch long strands. Set aside.
Pat the chicken dry. Slice each breast in half lengthwise. Transfer one of the chicken halves to a cutting board lined with parchment paper. Place another piece of parchment paper on top and use a mallet to pound the chicken to ¼-inch thickness. Continue until all 4 pieces of chicken have been pounded thin. Season the chicken with 1 teaspoon kosher salt and ½ teaspoon pepper. Transfer to the refrigerator. See Note 1.
Sauté the mushrooms:
Add the mushrooms to a skillet on the stove. Turn the heat to medium and cook until the water releases from the mushrooms and all the excess moisture evaporates, about 4 to 5 minutes. Once the water evaporates from the skillet, add 2 teaspoons extra virgin olive oil and sauté for 3 minutes until the mushrooms turn golden brown. Season with salt and pepper.
Add the garlic and thyme to the mushrooms and sauté over medium heat for 1 minute. Add the sweet vermouth and bring to a boil. Let the sweet vermouth bubble until it cooks down, about 2 to 3 minutes. Taste and season. Turn off the heat and transfer the mushrooms to a bowl. Wipe out the skillet.
Prepare the roulades:
Place a chicken cutlet on your work surface. Place 2 slices of Andrew & Everett Swiss Cheese on top. Spoon ¼ of the mushrooms on top of the cheese.
Starting at the end closest to you, carefully begin rolling the chicken around the mixture, tucking the mushrooms under as you roll. Once the chicken has been rolled up, use 2 pieces of twine to tie up both ends of the roulade. Trim the excess twine using kitchen shears. Set aside and prepare the remaining roulades.
Fry the roulades:
Heat 2 teaspoons of extra virgin olive oil in the skillet over medium heat. Add the roulades and cook for 2 to 3 minutes until browned.
Transfer the skillet to the preheated oven for 25 minutes or until the chicken reaches 165ºF. Remove from the oven.
To serve:
Remove the twine from the roulades and serve them with roasted vegetables or mashed potatoes. Enjoy!
Notes
Note 1: I recommend 1 teaspoon salt and ½ teaspoon pepper per pound of chicken. If your chicken weighs more than 1 pound, adjust the salt and pepper accordingly.