Heat a large skillet over medium-high heat. Cook and stir ground turkey in the hot skillet until browned and crumbly, 5 to 7 minutes. Remove from the heat. Drain and discard grease.
Melt butter in a skillet over medium heat; stir in carrots, onion, celery, and garlic. Cook and stir until the onion has softened and turned translucent, about 10 minutes.
Add broth, milk, parsley, thyme, and pepper. Stir in turkey and bring to a boil. Reduce heat to medium-low and simmer for 30 minutes. Stir in rice and cook until heated through, about 5 minutes.