Mince the garlic and quarter the mushrooms. Peel and cut the carrots into sticks. Chop the chorizo into small cubes. Strip the rosemary leaves off of the sprig. Remove the chorizo from the casing.
Pat the chicken dry with paper towel and season with salt and pepper. Heat 1 teaspoon olive oil in a pot over medium heat. Sear the chicken thighs for about 3 minutes on each side until golden brown and set aside. Add the chorizo and cook for a minute or two. Add the garlic and cook for 30 seconds until fragrant.
Add the tomatoes and mushrooms and nestle the chicken into the sauce. Season with salt and pepper. Lower the heat to low, cover, and simmer for about 15 minutes.
While the chicken cooks, place the potatoes into a pot with cold water. Add a generous pinch of salt. Bring to a boil and cook for 10-12 minutes. Add the carrots and cook for 3-4 minutes more until the potatoes are easily pierced with a knife and the carrots are crisp-tender. Drain.
Heat a non-stick pan over medium high heat. Add 2 teaspoons of olive oil and add the potatoes and rosemary. Cook, tossing occasionally, for a couple minutes until the potatoes are crispy. Add the carrots to the pan to re-heat. Season with salt and pepper.
Finish: Serve the chicken with the potatoes and carrots on the side. Pour some of the chorizo tomato sauce over the chicken.