Heat olive oil in a large skillet over medium heat. Add onion, carrots, celery, and garlic, cooking until softened, about 5 to 7 minutes.
Add ground turkey, thyme, rosemary, salt, and pepper. Cook until turkey is browned and cooked through.
Sprinkle flour over the turkey mixture and stir to combine. Add chicken broth and bring to a simmer. Stir in peas and corn. Cook until slightly thickened, about 3 to 5 minutes.
Meanwhile, boil the potatoes in salted water until tender, about 15 minutes. Drain and mash with milk and butter until smooth.
Spread the turkey mixture in a baking dish. Top with mashed potatoes and sprinkle with cheddar cheese.
Bake in the preheated oven for 35 minutes, or until the top is golden and bubbly.