Pat the chicken breasts dry with paper towels; sprinkle with kosher salt and freshly ground black pepper. Pound the chicken breasts gently with the smooth side of a meat tenderizer to an even thickness.
Heat 1 tablespoon of butter and 1 tablespoon of olive oil in a skillet over medium heat. Add the chicken breasts to the skillet and cook for about 6 to 8 minutes per side, depending on the thickness. The temperature of the chicken should be at least 165 F.
Remove the chicken to a plate.
Slice the green onions, separating the white and light green from the darker green tops. Put the tops in a small bowl and set aside. Put the white and light green parts in the skillet and cook until softened, stirring constantly. If necessary, add a small amount of oil or butter to the skillet. Add the garlic and cook for another minute. Add the wine and chicken stock and bring to a boil over high heat. Reduce the heat to medium-low and continue cooking until the liquids have reduced by half, about 4 to 6 minutes.
Stir the mustard, remaining 1 tablespoon of butter, and cream into the liquids and return the chicken to the skillet. Taste and adjust the seasonings. Heat through.