Position a rack in the middle of the oven and preheat to 450℉.
In a small bowl, whisk together the broth and Worcestershire sauce. In another small bowl, whisk together the salt, pepper and paprika.
Thoroughly pat the chicken dry. Separate the skin from the thigh meat by inserting your index finger between the two and lifting up gently.
Lightly brush a large rimmed baking sheet with oil. Place the chicken on it skin-side up and brush with 2 tablespoons of the olive oil. Sprinkle generously with the seasoning mixture, both on top and beneath the skin.
Pour the broth mixture around the chicken (not on top), being careful not to rinse off any of the seasoning. Roast the chicken for 20 minutes.
While the chicken is cooking, in a medium skillet, heat the remaining 1 tablespoon of olive oil and the butter over medium-heat until the butter has melted. Add the onions and sugar, season with a couple of generous pinches of salt, cover, and cook for 15 minutes, stirring every 5 minutes and adjusting the heat as needed to prevent the onions from burning. Remove the lid and cook until the onions turn a beautiful light caramel color, another 5 minutes or so. Stir in the thyme and cool until fragrant, about 2 minutes. Remove from the heat and set aside.
In a small bowl, combine the sherry, parsley and garlic. Remove the chicken from the oven and evenly pour the sherry mixture over the thighs. Evenly divide the caramelized onions over each thigh, followed by the Gruyere.
Return the chicken to the oven and bake for 15 minutes more, or until the cheese has fully melted. Let rest for about 5 minutes, garnish with the parsley, and then serve.