Use kitchen shears to cut off any excess skin from the chicken thighs. Finely chop the skin and refrigerate. Use a sharp paring knife to cut around the bone in each of the thighs and remove.
Whisk together the buttermilk, a pinch salt and pepper in a large bowl. Add the boneless chicken thighs and marinate in the refrigerator for at least 1 hour and up to 4 hours.
Once the chicken has marinated, make the gravy. Combine the chicken broth, whole milk and thyme in a small saucepan and bring to a simmer over medium heat. Turn off the heat and cover with a lid.
Meanwhile, put the olive oil and chicken skin in a large high-sided skillet. Turn the heat to medium and cook until the fat from the skin has rendered and the is skin is browned in spots, about 10 minutes, stirring occasionally. Add the onions, 1 teaspoon salt and a couple grinds of pepper and cook until tender, stirring frequently, about 10 minutes. Stir in 3 tablespoons of the flour and cook for 2 minutes, stirring constantly. Whisk in the milk mixture and bring the liquid up to a strong simmer, increasing the heat if needed. Cook until thickened to the consistency of gravy, about 5 minutes, stirring frequently. Season with salt and pepper if needed. Remove and discard the thyme. Cover with a lid to keep warm while you fry the chicken.
Fill a large Dutch oven a little less than halfway with vegetable oil and heat to 350 degrees F. Fit a baking sheet with a wire cooling rack.
Whisk together the remaining 1 1/2 cups flour, the cornstarch and Cajun seasoning in a medium bowl. One by one, remove the chicken thighs from the buttermilk marinade, allowing any excess to drip away, then dredge in the seasoned flour until completely coated.
When the oil is hot, fry the chicken in batches of 2 to 3 pieces until golden brown and the internal temperature is 165 degrees, 5 to 7 minutes total, carefully turning the chicken once halfway through the cooking time. Transfer the fried chicken to the cooling rack. Allow the oil to return to 350 degrees F in between each batch.
Serve immediately with the gravy. If the gravy thickens too much as it sits, whisk in additional chicken broth to thin the gravy out to your liking.