Place chicken between plastic wrap and pound to even 1/2-inch thickness. Pat dry thoroughly with paper towels - this is crucial for browning. Season both sides with 3/4 tsp salt and 1 tbsp cracked peppercorns, pressing the pepper into the meat so it adheres.

Heat olive oil and 1 tbsp butter in a large skillet over medium-high heat until shimmering. Add chicken in a single layer (work in batches if needed) and cook undisturbed for 4-5 minutes until deeply golden. Flip and cook 3-4 minutes more until internal temperature reaches 165°F. Transfer to a plate and tent loosely with foil.

Add remaining 1 tbsp butter to the same skillet. Add mushrooms in a single layer and cook without stirring for 3 minutes to develop color. Stir and continue cooking 4-5 minutes until browned and any released liquid has evaporated.

Add shallot and remaining 1/4 tsp salt. Cook 2 minutes until softened, then add garlic and cook 30 seconds until fragrant. Pour in brandy, scraping up all browned bits from the pan bottom. Let bubble and reduce by half, about 1 minute.

Stir in cream, Dijon, thyme, and remaining 1/2 tbsp cracked peppercorns. Simmer gently for 3-4 minutes until sauce coats the back of a spoon. Taste and adjust seasoning - the pepper should be prominent but not harsh.

Nestle chicken and any accumulated juices into the sauce, spooning mushrooms over top. Simmer 2 minutes to rewarm. Serve immediately with crusty bread, buttered noodles, or mashed potatoes to catch every drop of sauce.