Put the potatoes in a microwave-safe bowl and add 2 tablespoons water; cover and microwave until tender, 8 to 10 minutes. Let stand, covered, until ready to mash. Meanwhile, heat 1 tablespoon olive oil in a large skillet over medium-high heat. Add the onion, carrot, tomatoes, 3/4 teaspoon chili powder and 1/2 teaspoon salt; cover and cook, stirring occasionally, until the vegetables are tender, about 10 minutes. Transfer to a 2-quart baking dish.
Heat the remaining 1 tablespoon olive oil in the skillet over medium-high heat. Add the turkey, beef, the remaining 3/4 teaspoon chili powder and 1/2 teaspoon salt; cook, stirring and breaking up the meat with a wooden spoon, until browned, about 4 minutes. Stir in the peas and 1/4 cup water; simmer 2 minutes. Spread over the vegetables in the baking dish, then cover to keep warm.
Preheat the broiler. Add the milk and 1/4 teaspoon salt to the potatoes and mash well; mix in the cheese. Spread over the meat and broil until browned in spots, 1 to 2 minutes.