Place the chicken breasts in the bottom of the slow cooker.
Add the chopped carrots, celery, diced onion, and minced garlic on top of the chicken.
Pour the chicken broth over the ingredients in the slow cooker.
Sprinkle the dried thyme, dried parsley, salt, and black pepper evenly over the mixture.
Cover and cook on low for 4 hours, or until the chicken is cooked through and vegetables are tender.
Remove the chicken breasts from the slow cooker and shred them using two forks. Set aside.
In a small bowl, whisk together 1/4 cup flour and 1 cup whole milk until smooth to create a slurry.
In a small saucepan, melt 1 tablespoon butter over medium heat. Stir in the flour and milk slurry mixture, cooking and stirring until the mixture thickens, about 3-4 minutes.
Pour the thickened mixture into the slow cooker and stir to combine.
Return the shredded chicken to the slow cooker and stir well.
In a medium bowl, combine 1 cup flour, baking powder, and 1/2 teaspoon salt.
Add 1/2 cup whole milk and 2 tablespoons melted butter to the dry ingredients and stir until just combined to form the dumpling dough.
Drop spoonfuls (about 2 tablespoons each) of the dumpling dough onto the surface of the stew in the slow cooker.
Cover and cook on high for 30 minutes, or until the dumplings are cooked through and fluffy.
Serve hot and enjoy your slow cooker chicken and dumplings.