In a medium bowl combine spinach, eggs, ricotta cheese, 3/4 cup of the flour, the Asiago cheese and lemon zest; set aside (dough will be soft).
In a 4-quart Dutch oven, combine broth and thyme leaves. Bring to boiling.
In a shallow dish place the remaining 1/4 cup flour. Use teaspoons to scoop a portion of the dough into the flour. Use your fingers to gently roll dough in the flour to coat, forming an oval shape. Drop the coated dough pieces into the broth, 1/4 at a time. Stir gently once or twice to keep dumplings from sticking together. Simmer until dumplings float (1 to 1 1/2 minutes). Remove dumplings from Dutch oven with a slotted spoon; repeat with remaining dough.
Add carrot to broth; cook 5 minutes or until soft. Return all dumplings to the Dutch oven. Stir in chicken, heat through. Top with additional Asiago cheese, lemon zest, and/or thyme.