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1
Preheat broiler and butter an 8-inch square baking pan.
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2
On a rack of a broiler pan broil corn about 4 inches from heat, turning it occasionally, until lightly browned on all sides, about 10 minutes.
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3
Cool corn and cut kernels from cobs.
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4
In a large heavy saucepan stir together milk and cornmeal and add corn kernels and butter.
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5
Bring mixture to a boil and simmer, stirring constantly to keep mixture from sticking and burning and adding water, 1 tablespoon at a time, if mixture becomes too thick 20 minutes.
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6
Stir in cheese until incorporated and season with salt.
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7
Pour polenta into baking pan spreading evenly, and cool.
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8
Chill polenta until cold, 2 to 3 hours.
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9
Polenta may be made 2 days ahead and chilled covered.
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10
Preheat oven to 350F.
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11
If using fresh or dried shiitake discard stems.
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12
Slice mushrooms.
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13
In a 10-inch skillet heat butter over moderately high heat until foam subsides and saute mushrooms and shallots with salt to taste until mushrooms begin to brown, 5 to 6 minutes.
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14
Add broth and boil until reduced by about half.
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15
Remove skillet from heat and add thyme, pepper, and salt to taste.
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16
Keep mushrooms warm.
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17
Unmold polenta onto a cutting board and cut into 16 triangles or 2-inch squares.
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18
On a lightly oiled baking sheet bake polenta pieces until heated through, 10 to 15 minutes, and broil about 4 inches from heat until golden, about 2 minutes.
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19
Arrange 2 polenta pieces on each of 8 plates.
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20
Spoon mushrooms over polenta and garnish with nuts.
