Sprinkle chicken with salt and pepper. Melt 1 tablespoon butter with 1 tablespoon oil in large skillet over medium-high heat. Add chicken to skillet and cook until browned and cooked through, 5 to 6 minutes per side, depending on thickness. Transfer chicken to plate; cover to keep warm. Add shallot to same skillet; stir over medium-high heat 1 minute. Add wine, then broth to skillet; bring to boil. Whisk in mustard and chopped tarragon; boil until sauce thickens enough to coat spoon lightly and is reduced to scant 1/2 cup, 5 to 6 minutes. Stir in 1/2 tablespoon butter; set aside.

Meanwhile, melt remaining 1 tablespoon butter with remaining 1 tablespoon oil in heavy medium skillet over medium-high heat. Add radishes; sprinkle with coarse salt and pepper; cook without stirring until radishes begin to brown, about 4 minutes. Continue to cook radishes until crisp-tender, stirring occasionally, 5 to 7 minutes longer.

Cut chicken breast halves crosswise into 1/2-inch-thick slices; arrange on platter. Place radishes around and over chicken. Spoon sauce over chicken and radishes. Garnish with tarragon sprigs and serve.