In a medium saucepan, heat 2 teaspoons of the oil over medium-low heat. In a medium bowl, beat eggs with 1/4 teaspoon coarse salt and 1 tablespoon water until combined. Add eggs to pan and cook, pulling eggs in from edges as they set, until center is just set, about 3 minutes. Turn out onto a plate. When cool enough to handle, halve, then cut each half into 1/4-inch-wide strips.
In the same pan, heat the remaining teaspoon of oil over medium heat. Add the scallions and garlic and cook, stirring, until fragrant, about 30 seconds. Add the broth, 1/2 cup water, 1/4 teaspoon each salt and pepper, and the ginger. Bring to a boil. Add the spinach and cook until the spinach has wilted and is tender, but not mushy, about 3 minutes. Stir in the egg strips. Remove from heat; correct seasoning with salt and pepper if desired.