Trim excess fat from chicken breast, and slice chicken in half lengthwise (the best method is the place your palm above the breast, and gently glide a sharp knife through the center of the breast parallel to your hand with gentle sawing motions until you have 2 even pieces per breast).
Pound each of the 4 chicken cutlets to ¼” thickness between plastic wrap or in a resealable bag, using a meat mallet or rolling pin. Be gentle to avoid tearing the chicken cutlets.
Season both sides of each chicken cutlet with salt and pepper.
Heat olive oil in a large deep nonstick skillet or cast iron skillet over medium heat for 2-3 minutes, then add pancetta.
Cook pancetta, stirring occasionally, until fat is rendered and pancetta is crisp, 5-7 minutes.
Using a slotted spoon, remove pancetta to a paper towel lined plate, and leave rendered fat in the pan.
Add the seasoned chicken to the pan and cook 3-4 minutes a side, until well browned, and transfer to a plate. Work in 2 batches to ensure even browning and avoid overcrowding the pan (Chicken will not yet be cooked through).
Reduce heat to medium-low, and add the diced shallot. Cook, stirring occasionally for 2-3 minutes, until slightly translucent.
Add the garlic and cook until just fragrant, 30 seconds.
Deglaze the pan with the wine or vermouth, scraping up any brown bits.
Cook for 2-3 minutes, until most of the wine is evaporated.
Add the chicken stock, cream, thyme and pancetta and stir well.
Add the chicken and accumulated juices to the pan.
Simmer over medium heat for 10 minutes, until sauce is slightly thickened, and the chicken reaches 165°F internally.
Turn off heat and stir in butter.
Serve warm, and remove thyme before serving.