1. Preheat oven to 450. Place potatoes in a large saucepan, and cover by 1 inch with salted water. Bring to a boil; reduce to a simmer. Cook until potatoes are easily pierced with the tip of a paring knife, 15 to 20 minutes.

2. Meanwhile, heat oil in a 5-quart Dutch oven or heavy pot over medium-high. Add carrots, celery, onion, and thyme. Cook, stirring occasionally, until vegetables are tender, 8 to 10 minutes. Add flour and tomato paste; cook, stirring, 1 minute. Add beef; cook, stirring occasionally, until no longer pink, 6 to 8 minutes. Add 1 cup water; bring to a boil, and simmer 1 minute. Set beef filling aside.

3. Drain potatoes; return to pan. Cook over medium, stirring, until liquid has evaporated and a thin film covers bottom of pan, about 1 minute. Remove pan from heat; add milk and 1 cup cheese. Mash until smooth; season cheddar-potato topping with salt and pepper.

4. Pour beef filling into a 13-by-9-inch baking dish. Drop dollops of topping over filling; spread to edges with a spatula. Using a fork, make decorative peaks; sprinkle with remaining 1/2 cup cheese. Bake until topping is browned and filling is bubbling rapidly, about 20 minutes (if topping and filling were chilled, increase to 35 minutes). Let stand 5 minutes before serving.