Season the chicken breast lightly with salt and pepper.
In a medium saucepan, bring the low-sodium chicken broth to a simmer. Add the chicken breast and poach over medium-low heat for about 15 minutes or until fully cooked.
While the chicken is cooking, prepare the dumpling dough. In a small bowl, combine the whole wheat flour with 1/4 cup of water and a pinch of salt. Mix until a smooth dough forms, adding a bit more water if necessary.
On a lightly floured surface, roll the dough out to about 1/8-inch thickness. Cut the dough into small squares (approximately 3 inches per side).
Mix the fresh spinach (finely chopped) and grated carrot together in a bowl to serve as the dumpling filling.
Place a small spoonful of the spinach and carrot mixture onto each dough square. Moisten the edges with a little water, fold over to create a dumpling shape, and press to seal securely.
When the chicken is nearly done, carefully add the dumplings into the simmering broth, nestling them around the chicken. Continue to simmer for another 6-8 minutes until the dumplings are tender and cooked through.
Remove the chicken and dumplings from the broth, slice the chicken if desired, and serve with a few spoonfuls of the flavorful broth.