Place cubed potatoes in a large pot of salted water, bring to a boil, and cook for 15 minutes or until tender.
Preheat oven to 400°F (200°C).
Heat olive oil in a large skillet over medium-high heat. Add chicken and brown for 5-7 minutes; remove chicken and set aside.
In the same skillet, add onion, carrots, and celery. Sauté for 6 minutes until vegetables are softened.
Stir in minced garlic and cook for 1 minute. Sprinkle flour over the vegetables and stir constantly for 2 minutes to cook out the raw flour taste.
Gradually whisk in chicken broth and thyme. Bring to a simmer and stir until the sauce has thickened.
Return chicken to the skillet and stir in the frozen peas. Season with salt and pepper to taste.
Drain the cooked potatoes and mash with butter and milk until smooth.
Spread the mashed potatoes evenly over the chicken filling in a 9x13 inch baking dish.
Sprinkle shredded cheddar cheese over the potato layer.
Bake for 20-25 minutes until the potato topping is golden brown and the edges are bubbling.