Pat chicken breasts completely dry with paper towels, then slice each horizontally through the middle to make 4 thin cutlets. Pound any uneven cutlets to about ½ inch (1.25 cm) thick, season both sides with ½ teaspoon kosher salt, and set aside.
Crush 2 tablespoons whole black peppercorns in a mortar and pestle (or zip-top bag with a rolling pin) until coarse and cracked, not finely ground. Divide in half and press 1 tablespoon firmly onto both sides of each cutlet.
Heat a 12-inch skillet over medium-high heat, then add 1 tablespoon olive oil and 1 tablespoon butter. Sear the cutlets in a single layer for about 3 minutes per side until deep golden and the thickest part reads about 155°F (68°C). Transfer to a clean plate and do not wash the pan.
Reduce heat to medium, add 1 tablespoon butter to the skillet, and cook the minced shallot for 1 to 2 minutes until soft and translucent. Add the minced garlic and 3 thyme sprigs and stir for about 30 seconds until fragrant, without browning the garlic.
Take the skillet off the heat, carefully pour in ¼ cup cognac or brandy, then return to medium-high heat. Let it bubble for about 1 minute, scraping up the browned bits, until the liquid reduces by half and the sharp alcohol smell fades.
Pour in ½ cup low-sodium chicken broth, bring to a steady simmer, and cook for 3 to 4 minutes until reduced by about half and slightly glossy.
Lower heat to medium-low and stir in ¾ cup heavy cream, 1 teaspoon Dijon mustard, the remaining 1 tablespoon crushed peppercorns, and the remaining ½ teaspoon kosher salt. Simmer gently for 4 to 5 minutes, stirring often, until the sauce coats the back of a spoon.
Return the cutlets and any collected juices to the skillet, spoon sauce over the top, and simmer gently for 2 to 3 minutes until the chicken reaches 165°F (74°C). Discard the thyme stems, then swirl in the remaining 1 tablespoon butter off the heat and stir in 1 tablespoon chopped parsley.
Transfer the cutlets to warm plates and spoon plenty of sauce over each one. Sprinkle with extra chopped parsley and serve with lemon wedges on the side.