Melt butter in a large skillet over medium heat; add shallot, carrot, garlic, and 1/2 teaspoon kosher salt. Cook and stir until tender, 1 to 2 minutes. Remove from heat and set aside to cool.

Preheat oven to 425˚F. Line a baking sheet with parchment paper (for easy cleanup) and place an oven safe baking rack on top. Lightly brush rack with oil to prevent sticking. (Recipe note #2)

Carefully remove skin from chicken thighs in one piece, using a sharp paring knife as needed and reserve. (Recipe note #3)

Turn chicken thigh so that bone is visible. With the tip of a sharp paring knife, carefully cut along the length of the bone on each side. Continue cutting along the bone on each side until it's fully exposed. When you've cut most of the meat from the bone, pull the bone away and use the knife as needed to remove it completely. (Be very careful to protect your fingers during this process!) Cut thigh meat into large chunks (4 to 6 pieces). Repeat with remaining chicken thighs.

Add thigh meat, vegetables, remaining 1/2 teaspoon kosher salt, and pepper to the bowl of a food processor. Pulse until meat is finely ground (not pureed) and ingredients well combined.

Spread thigh skins, top side down on a large cutting board. Divide meat mixture into 4 equal portions and place one portion in center of each skin. Starting on one side and working all the way around, pull edges of skin over meat mixture to wrap completely (edges will overlap); place on rack in baking sheet, edges down. Season skin with salt and pepper. Roast 25 to 30 minutes, or until skin is crisp and golden brown with an interior temperature of 165˚F on an instant read thermometer. Serve.