Preheat oven to 350°. Place the chicken on a sheet pan, skin side up. Loosen a pocket between the skin and breast meat with two fingers lightly under the skin. Grind the chouriço with a few pulses in the food processor or finely chop until somewhat smooth.
Gently insert 2–3 tablespoons chouriço in the pocket you created underneath the chicken skin and smooth out by pressing gently to remove lumps. Once complete, season chicken with salt and pepper.
Heat 2 tablespoons oil in a large ovenproof sauté pan over medium-high heat. Sear, skin side down, until the skin is amber in color, then flip over. (If chicken is too crowded for browning, do this step in batches.)
Add 3 tablespoons butter to the pan and allow to melt. Baste the chicken with the butter mixture and place in the oven until the internal temperature reaches 165° (about 25 minutes).
Meanwhile, add the carrots, ¾ cup chicken stock and cayenne pepper to a saucepot, bring to a boil, then simmer until the carrots are tender. Add a little water if the liquid is reducing faster than the carrots cook. When tender, remove from heat. In a blender, purée carrots and liquid along with 2 tablespoons butter and lemon juice until smooth. Season to taste and keep warm.
When chicken is done, remove from oven. Set aside on a plate and allow to rest in a warm spot for 5–6 minutes while you make the pan sauce.
Reserving the pan juices, and being careful of the hot handle, place the pan on medium-high heat and add 1 cup chicken stock. Reduce the liquid by ⅓ on high heat. Add the thyme, 1 tablespoon butter and stir until the sauce starts to thicken. Taste first then add salt, if needed. Keep warm.
Lastly (or while carrots and chicken are cooking), heat 1 tablespoon neutral oil in a sauté pan over medium high heat. Add the snow peas, toss and shut the heat off. Allow the heat of the pan to cook the snow peas but retain their crunch.
To serve, spoon a circle of carrot purée on each of 6 warm dinner plates. Add chicken, snap peas (and optional gnocchi as directed). Drizzle with pan sauce and serve. Enjoy!