Bring a large pot of lightly salted water to a boil. Add tortellini and cook 7 minutes (or as per package directions).
Meanwhile, dice chicken into 1-inch pieces. Drain tortellini, reserving 1/4 cup of the cooking water.
Heat oil in a large skillet over mediumhigh heat. Sprinkle chicken with 1/8 tsp of the salt. Add to skillet and brown on all sides, 5 minutes. Stir in spinach, tomatoes, heavy cream and remaining 1/2 tsp salt. Simmer 1 minute. Add tortellini and reserved pasta water, if needed; cook 2 minutes. Remove from heat and toss with Parmesan. Grind black pepper over top.