Once the chicken is cooked through, remove the pot from the oven. Transfer the chicken to a large plate and return the pot to the stove over medium heat. Use a spoon to skim off excess chicken fat from the pot if needed.
Add 3 tablespoons tomato paste and a pinch of crushed red pepper, and stir to coat the vegetables. Cook for 1 to 2 minutes, until the tomato paste emulsifies with the liquid in the pot.
Add the reserved pasta water and bring to a boil. Reduce the heat to medium-low and simmer, uncovered, for 5 to 10 minutes until reduced and thickened. Taste and season with salt, pepper, crushed red pepper, or a pinch more sugar if needed.
Add the cooked linguine to the sauce and toss to coat. Turn off the heat.