Pat the chicken dry, then coat it with 1 teaspoon salt, 1 teaspoon pepper, and the flour.
Melt the butter in a large, deep skillet over medium heat.
Add the chicken and brown on either side, about 4 minutes per side.
Add the onion and garlic and stir until fragrant.
Add the carrot, celery, and 2 cups of chicken broth and bring to a simmer. Simmer until the chicken is cooked through, about 10 minutes further.
Remove the chicken and shred it using two forks.
While the chicken is simmering, make the dumpling batter. In a large bowl, mix the flour, milk, melted butter, ½ teaspoon salt, ½ teaspoon pepper, rosemary, and thyme together until mostly smooth.
Return the chicken to the skillet with the heavy cream and the remaining cup of broth and stir.
Bring the liquid to a simmer over medium-low heat and drop in the dumplings using a cookie scoop.
Cover and steam until the dumplings are cooked on the bottom, about 5 minutes. Remove the lid and turn each dumpling over.
Return the lid and cook until the dumplings are solid and cooked through, about 10 minutes. The broth will be thick.
Once the dumplings are fluffy throughout, serve the chicken and dumplings.