Cook potatoes in boiling water until tender; drain and mash. Add the garlic, milk, butter, rosemary, salt and pepper; set aside.
In a large skillet, saute mushrooms, carrots, onion and green pepper in butter until tender. Add the venison, broth, thyme and nutmeg. Bring to a boil. Reduce heat; cover and simmer for 25-30 minutes or until meat and vegetables are tender.
Combine the flour and water until smooth; stir into skillet. Bring to a boil; cook and stir for 2 minutes or until thickened. Transfer to a greased 2-qt. baking dish. Spread mashed potatoes over the top.
Bake, uncovered, at 350° for 30-40 minutes or until bubbly. Sprinkle with cheese. Bake 5 minutes longer or until cheese is melted. Sprinkle with parsley.