Heat a large Dutch oven over medium-high. Add oil to pan; swirl to coat. Add onion and carrot; sauté 4 minutes. Add garlic and tomato paste; cook 2 minutes, stirring occasionally. Add stock, 1 cup water, and salt; bring to a boil. Add pasta to pan; cook 5 minutes. Add snap peas, tomato, and chicken to pan; cook 3 minutes or until pasta is done.