Brush both sides of chicken breasts with dijon mustard. Season generously with salt and pepper.
Heat 1 tablespoon olive oil in a skillet over medium-high heat. Cook the chicken breasts until they reach an internal temperature of 165 degrees, flipping half way through. This should take about 6 to 8 minutes if using thin chicken breasts.
Remove the cooked chicken from the skillet and set aside.
Using the same skillet, pour in the remaining 1 tablespoon olive oil. Add in mushrooms and shallots. Saute for 3 to 5 minutes. Add in garlic and cook 1 minute more.
Pour in heavy cream and tarragon. Bring to a simmer and cook until the sauce has thickened, about 5 minutes.
Return the chicken to the sauce and let them heat through. Serve hot over cooked pasta.