Melt the butter in a large pot over medium heat. Add the carrots, celery, and onion along with 1/2 teaspoon of salt. Cook until the onions are translucent. Add the water and chicken base and stir until the chicken base dissolves.

In a mixing bowl, combine the flour, baking powder, buttermilk, and 1/2 teaspoon of salt. Stir until the dough just comes together—it will be shaggy.

Transfer the dough to a floured surface and roll it out into a rectangle about 1/4 inch thick. Cut the dough into 1-inch squares, separating the squares as you cut them so they don’t touch one another. Sprinkle the squares with more flour.

Strain the vegetables from the broth and return only the broth to the pot (discard the vegetables). Bring to a boil. Slowly add the dumplings, a few at a time, stirring as you go.

Reduce the heat to medium-low and simmer for 15 minutes to ensure the dumplings are cooked through. Taste and adjust the seasonings.