3 teaspoons extra-virgin olive oil, or canola oil, divided
1/4 cup finely chopped onion
3 cloves garlic, minced
1 cup reduced-sodium chicken broth
2 teaspoons flour
2 tablespoons chopped fresh dill, divided
1 tablespoon lemon juice
Season chicken breasts on both sides with salt and pepper. Heat 1 1/2 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 1 1/2 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. Transfer the chicken to a warmed platter. Season sauce with salt and pepper and spoon over the chicken. Garnish with the remaining 1 tablespoon chopped fresh dill.