Preheat oven to 350 degrees F (175 degrees C).

Place potatoes into a large pot and cover with water; bring to a boil. Cook potatoes until partially cooked, about 4 minutes; drain.

Mix flour, 1/2 teaspoon salt, and 1/2 teaspoon pepper together in a shallow bowl. Press chicken into flour mixture until evenly coated.

Melt butter in a large skillet over medium heat; place chicken in hot butter and sprinkle 1 tablespoon tarragon over chicken. Cook chicken until browned, 3 to 4 minutes per side.

Arrange 1/2 of the potatoes in a baking dish and top with chicken breasts. Layer remaining potatoes over chicken. Mix cream, garlic, 1 teaspoon salt, and 1 teaspoon pepper in a bowl; pour over potatoes and chicken. Sprinkle remaining tarragon over potatoes.

Bake in the preheated oven until chicken is until no longer pink in the center and the top is golden brown, 45 to 50 minutes. An instant-read thermometer inserted into the center should read at least 165 degrees F (74 degrees C). Let stand for at least 10 minutes before serving.