Heat oven to 400°F. Spread 1 tablespoon butter all over a 2 ½ to 3-quart baking dish and set aside.
Combine cream, shallot, garlic, thyme, rosemary, 1 ½ teaspoons salt and ½ teaspoon black pepper in a small saucepan over medium-low heat. Bring to a gentle simmer and cook 15 minutes, or until flavorful and thickened. Set aside to cool slightly.
Spread ⅓ of the carrots and potatoes in the dish and top with ¼ cup grated parmesan. Repeat layering vegetables, ¼ cup parmesan and vegetables. Strain warm cream over the top; discard solids.
Dot the surface with remaining 2 tablespoons butter and sprinkle with remaining ¼ cup parmesan. Cover with foil and bake for 45 minutes, then remove foil and bake another 25-30 minutes, or until vegetables are tender, cream has thickened and top is golden brown. Let stand 10 minutes before serving.
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