Preheat the oven to 450 degrees F (230 degrees C).
Heat 1 tablespoon oil in a large skillet or Dutch oven over medium-high heat. Add chicken; cook, turning occasionally, until browned on all sides, 8 to 10 minutes. Transfer chicken to a large bowl.
Reduce heat to medium. Add 1 tablespoon oil, celery, carrot, and onion to the skillet. Cook, stirring occasionally, until vegetables are tender, about 10 minutes. Add garlic; cook until fragrant, about 1 minute more.
Return chicken to skillet along with stock, butter, whipping cream, thyme, salt, and pepper. Bring to a simmer over medium-high heat, then stir in potato flakes. Cook and stir until thickened and bubbly. Stir in parsley.
Put tots in a large microwave-safe bowl. Microwave on high, stirring once per minute, until defrosted and broken up, about 5 minutes. Stir in remaining 1/4 cup oil until incorporated.
Set six 14- to 16-oz individual baking dishes on a large baking sheet. Divide chicken mixture among dishes. Top each with tots mixture, pressing down gently to flatten into a crust.
Bake in the preheated oven until filling is bubbly at the edges, about 30 minutes.
Preheat broiler to high and broil until crusts are browned and extra crisp, about 1 minute. Watch carefully to avoid burning. Let cool slightly before serving.
