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

Add the water and chicken base to the pot. Stir until the chicken base is incorporated into the water. Add the chicken tenders and cook for about 20 minutes on medium-low heat.

While the chicken is cooking, prepare the dough for the dumplings: Combine the flour, baking powder, buttermilk, and 1/2 teaspoon of salt in a mixing bowl. 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 do not touch one another. Sprinkle the squares with more flour.

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

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