In a large skillet over medium-high heat, heat oil. Season chicken on all sides with ½ tsp. salt and ½ tsp. pepper. Cook chicken, turning halfway through, until golden brown, 7 to 8 minutes per side. Transfer to a plate.
Reduce heat to medium and melt butter. Add shallot and garlic and cook, stirring, just until fragrant, about 2 minutes. Add broth, mustard, horseradish, paprika, chives, parsley, and thyme, and whisk to combine; season with remaining ½ tsp. salt and ½ tsp. pepper.
Bring mixture to a low simmer. Return chicken and any accumulated juices to skillet. Cook, stirring occasionally, until an instant-read thermometer inserted into thickest part of chicken registers 165° and sauce is slightly reduced, 8 to 10 minutes, depending on thickness of chicken. If liquid reduces too much, add 1 to 2 Tbsp. broth to loosen into a pourable sauce.
Serve chicken with sauce poured over. Sprinkle with chives.