Pat chicken dry with a paper towel. Season the breasts with salt and pepper.
Heat vegetable oil in a stainless steel skillet over medium high heat. Add chicken skin side down and cook until golden brown, about 5 minutes. Flip chicken and repeat on other side. Move chicken to a plate and pour out all but 2 teaspoons of oil.
Return pan to medium heat. Add shallots and cook, stirring occasionally, until softened and translucent.
Add wine and simmer until reduced to about 2 tablespoons.
Whisk in mustard, then chicken broth. Return chicken to pan and simmer on low heat until chicken is cooked through, about 10 minutes. Remove the chicken from the pan and set aside once more.
When the liquid has reduced to about 1/4 cup, whisk in cold butter until completely melted. Squeeze lemon over sauce and add chopped herbs.
Return chicken to pan and turn to coat in sauce. Serve with remaining sauce.