Thinly slice scallions and set aside dark green parts. Drain tofu and cut into half-inch cubes.
In a medium saucepan with a tight-fitting lid, mix the broth with the scallion slices and the spinach. Cook, covered, over med-high heat, removing the lid a few times to break up the spinach with a wooden spoon, until thawed.
Bring soup to a boil, about ten minutes. Reduce heat to med-low, add tofu without stirring and cook until heated through, 2-3 minutes. Remove pan from heat, add dark green scallion slices and sesame oil and season to taste with salt and pepper.