Season chicken breasts on both sides with salt and pepper. Heat 1½ teaspoons oil in a large heavy skillet over medium-high heat. Add the chicken and sear until well browned on both sides, about 3 minutes per side. Transfer chicken to a plate and tent with foil.

Reduce heat to medium. Add the remaining 1½ teaspoons oil to the pan. Add onion and garlic and cook, stirring, for 1 minute. Whisk broth, flour, 1 tablespoon dill and lemon juice in a measuring cup and add to pan. Cook, whisking, until slightly thickened, about 3 minutes.

Return the chicken and any accumulated juices to the pan; reduce heat to low and simmer until the chicken is cooked through, about 4 minutes.

Season sauce with salt and pepper and spoon over the chicken. Garnish with the remaining 1 tablespoon chopped fresh dill.