Preheat an outdoor grill for medium-high heat. Preheat oven to 160 degrees F (71 degrees C).
Spread 1 tablespoon olive oil and 1/2 teaspoon salt over the chicken pieces.
Place chicken pieces skin-side down on the hot grill; cook until browned, turning occasionally, about 20 minutes for the breasts and 30 minutes for the leg pieces. An instant-read thermometer inserted near the bone should read 165 degrees F (74 degrees C). Transfer to a baking sheet and keep warm in the preheated oven.
Combine remaining 1 1/2 tablespoon olive oil, remaining 1/4 teaspoon salt, butter, tarragon, and lemon juice in a food processor; blend until smooth.
Divide grilled chicken pieces among 4 serving plates. Dot each serving with about 1 tablespoon of the tarragon butter.
