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1
Whisk together water, cornmeal, and salt in a 1-quart heavy saucepan and bring to a boil over moderate heat, whisking constantly.
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2
Reduce heat to low and cook, stirring frequently, until very thick, 15 to 20 minutes.
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3
Spread polenta into a 10- by 5-inch rectangle on a lightly oiled baking sheet.
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4
Chill, uncovered, until firm, about 1 hour.
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5
Prepare grill for cooking.
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6
Cut 4 (1/4-inch-thick) crosswise slices from middle of each tomato.
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7
Sandwich slices between triple thicknesses of paper towel and press gently to absorb excess moisture.
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8
Preheat oven to 400F.
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9
Cut out 8 (2 1/2-inch) rounds from polenta with cookie cutter.
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10
When fire is medium-hot (you can hold your hand 5 inches above rack for 3 to 4 seconds), grill polenta rounds in a grill basket until crusty, about 5 minutes on each side.
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11
Transfer to an ungreased baking sheet.
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12
Season tomato slices with salt and pepper and sprinkle with cheese.
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13
Assemble 4 stacks by placing a tomato slice on top of each of 4 polenta rounds (use a spatula to transfer tomato slices), then top each with another polenta round and tomato slice.
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14
Bake in middle of oven until hot and cheese is melted, about 15 minutes.
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15
Puree basil, water, garlic, lemon juice, oil, and sugar in a blender, scraping down sides frequently, until smooth.
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16
Season with salt and pepper.
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17
Serve stacks with basil sauce.
