Heat oil in a large skillet over medium-high heat. Add chicken; cook until well browned and no longer pink in the center, 4 to 5 minutes per side. Transfer to a plate, cover, and keep warm.
Add mushrooms and leeks to the skillet. Cook over medium heat, stirring often, until most of the moisture has evaporated and mushrooms begin to brown, 4 to 6 minutes. Add wine; cook for about 1 minute. Mix broth and cornstarch together in a small bowl; add to the skillet and cook until thickened, 2 to 3 minutes. Stir in sour cream and mustard; cook and stir until combined, about 1 minute. Season sauce with salt and pepper.
Nestle chicken thighs into the sauce. Cook until an instant-read thermometer inserted into the thickest part of a thigh reads 165 degrees F (74 degrees C), 8 to 10 minutes more.