Stir together sour cream, mayonnaise, Dijon mustard, capers, caper brine, dill weed, and 1/4 teaspoon black pepper in a small bowl. Stir in 1 teaspoon lemon juice. Taste; add lemon juice as desired, up to 2 teaspoons more, reserving remaining lemon zest and juice for chicken. Refrigerate sauce until ready to use.

Combine flour, 1 teaspoon lemon zest, garlic powder, salt, and 1/2 teaspoon black pepper in a resealable plastic bag. Shake to combine.

Pat chicken dry with paper towels; add to the bag. Squeeze out most of the air, reseal the bag, and move chicken around until lightly coated with flour mixture.

Heat butter and oil in a 12-inch skillet over medium heat; swirl the skillet a few times to mix. Add chicken to the hot skillet in a single layer; cook until browned, 3 to 4 minutes. Turn; cook until chicken is browned, no longer pink at the center, and juices run clear, 3 to 4 minutes. Internal temperature measured with an instant-read thermometer should be at least 165 degrees F (74 degrees C). Remove chicken to a plate; keep warm.

To the same skillet, add asparagus; cook and stir until bright green, about 2 minutes. Remove to a plate; keep warm. Stir minced garlic into skillet; cook until fragrant, about 30 seconds. Pour chicken broth and 2 tablespoons lemon juice into the skillet; bring to a boil, scraping up the browned bits from the bottom of the pan.

Return chicken to the skillet, and turn chicken to coat both sides in the liquid; return to a boil. Return asparagus to the skillet, and continue cooking until most of the liquid has evaporated, 1 to 2 minutes.

Place chicken and asparagus on a serving plate and add a dollop of Lemon-Caper Cream.