Mix in the tomatoes to deglaze the pan. Add the potatoes, carrots, celery, broth, remaining 1 teaspoon salt, and a few grinds of black pepper. Cover and bring to a boil. Once boiling, lower the heat and simmer, covered, until the potatoes are tender, about 15 minutes. Lower the heat to the lowest setting and cook, covered, for 20 more minutes to develop the flavors.