On a large cutting board, place chicken between sheets of wax paper. Pound with a meat mallet, rolling pin or heavy skillet until meat is an even 1/2 inch in thickness. Season chicken with 1/2 tsp salt (1/4 tsp on each side).
Heat a large skillet over medium-high heat. Coat with oil, then add butter. When butter is foamy but not brown, add half the chicken. Cook until browned, about 3 minutes. Flip chicken and cook until browned but not cooked through, about 2 additional minutes. Transfer to a plate. Repeat with remaining chicken.
Add onion to skillet. Cook 1 minute, scraping brown bits from pan. Add wine and cook until reduced by half, about 1 minute, then add stock. Simmer 1 minute.
Whisk in mustard and cook 1 minute. Stir in cream and remaining 1/8 tsp salt. Bring to a boil; reduce heat to a simmer, then add chicken back to pan with its juices. Simmer until chicken is cooked through, about 2 minutes. Stir in thyme and tarragon.