In a medium pot, bring the chicken broth to a simmer over medium-high. Add the cabbage and cook, stirring occasionally, until the cabbage is just tender, 3 to 5 minutes. In a small bowl, whisk together the egg and cheese.
Add the pastina and cook, stirring often, until the pasta is just tender, 5 to 6 minutes. Remove the pot from the heat and stir in butter.
Stir a few spoonfuls of the hot pasta mixture into the cheese mixture. Repeat this process 2 more times, then return everything to the pot. Stir in the lemon zest and season to taste with salt and pepper. Serve immediately.
