Melt butter in a large Dutch oven over medium-high heat. Add diced onion, sliced carrots, and chopped celery, stirring occasionally for 5-7 minutes until softened and fragrant.
Place chicken pieces into the vegetable mixture, then pour chicken broth over everything until mostly covered. Add dried thyme, salt, and black pepper, then bring to a gentle boil for 2 minutes before reducing heat.
Simmer the creamy hearty chicken and dumplings base for 15-18 minutes until chicken reaches 165°F internally, then carefully remove pieces and shred the meat while broth continues simmering gently.
Whisk together flour, baking powder, and salt in a medium bowl, then create a well in the center. Pour milk and melted butter into the well, stirring until just combined—do not overmix or dumplings become tough.
Return shredded chicken to the pot, then stir in heavy cream and fresh parsley. Drop dumpling batter by the spoonful directly onto the simmering surface, allowing 12-15 minutes for dumplings to cook through without stirring.