Preheat oven to 400° with a rack in top third of oven and a second one below it.
Trim ends from asparagus, then toss with 1 tbsp. olive oil on a rimmed baking sheet. Sprinkle with 1/4 tsp. each salt and pepper. Roast on lower rack until tender and lightly charred, 15 to 20 minutes.
Meanwhile, stir together remaining 1/2 tsp. each salt and pepper with the paprika and garlic powder. Rub chicken with 1 tbsp. oil and the spice rub.
Heat remaining 2 tbsp. oil in a large ovenproof pan over medium-high heat. Add chicken; cook until browned beneath, 4 minutes. Turn over, then bake on upper rack until cooked through, 8 to 10 minutes. Transfer to a platter; tent with foil.
Discard all but 11/2 tbsp. drippings from pan. Add shallot and cook over medium heat, stirring, until softened, 2 minutes. Increase heat to high, stir in wine and chopped tarragon; boil until liquid reduces by half. Add broth and juices from cooked chicken; boil until reduced to 1/3 cup, 10 minutes. Stir in lemon juice, remove from heat, and stir in butter. Set asparagus next to chicken. Pour sauce over chicken and sprinkle with tarragon leaves.