Rub chicken thighs with 1 tablespoon olive oil and salt. Let sit for 5 minutes, and set the oven to preheat to 400 degrees F.
Bring remaining 1 tablespoon olive oil to medium high heat in a 12 inch cast-iron skillet. Wait until very hot, then add chicken thighs skin side down. Cook for 3 minutes, then turn heat down to medium and continue to cook for 5 more minutes so that it can “fry” in its own fat, until skin is golden brown. Flip thighs and cook for another 8 minutes. Brush chicken with it's own juices, then transfer skillet to oven. Cook for 8-10 minutes longer, until meat is cooked through and juices run clear, registering 160 degrees F.
While chicken thighs are in the oven, make the glaze. Heat olive oil to medium heat in a small saucepan. Add shallot and salt, cooking for 3-4 minutes until translucent. Stir in mustard, vinegar and preserves. Lightly simmer for 5-6 minutes, until slightly thickened. Remove from heat and stir in basil.
When chicken thighs have reached 160 degrees F, transfer to a plate, smother with glaze, and let rest for 5 minutes before serving.