Pat thighs dry and season with 1/2 teaspoon salt and the pepper. Heat oil in a 12-inch skillet over medium-high heat.
Cook thighs unfolded for 6 to 7 minutes on the first side and 5 to 7 minutes on the second until browned and at least 165°F in the thickest part. Transfer to a plate.
Reduce heat to medium. Add asparagus and remaining salt to the skillet and cook for 4 minutes, turning, until bright green and lightly browned.
Add butter and garlic and stir for 1 minute. Add broth, cover and cook for 2 minutes until the asparagus is tender; stir in lemon juice.
Return chicken and its juices and turn in the garlic butter for 1 minute. Serve each thigh with asparagus and a little pan sauce.