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1
Sprinkle eggplant with 1 1/2 teaspoons salt in a colander and let drain in sink, turning slices occasionally, 30 minutes.
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2
Finely chop together basil, mint, and garlic.
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3
Toss with tomatoes in a bowl, then drain in a sieve set over bowl.
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4
Preheat broiler.
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5
Rinse eggplant, then pat dry in batches between paper towels, pressing firmly to remove excess liquid.
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6
Brush both sides of slices with oil (about 4 tablespoons total) and form small stacks, then cut stacks into quarters.
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7
Arrange eggplant pieces in 1 layer in 2 shallow heavy baking pans and broil, 1 pan at a time, about 5 to 6 inches from heat, turning eggplant over once, until golden brown, 7 to 8 minutes per batch.
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8
Cool eggplant to room temperature.
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9
Put pizza stone in lower third of oven and preheat oven to 500F (allow 1 hour for pizza stone to heat).
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10
Do not punch down dough.
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11
Gently dredge 1 piece of dough in flour to coat, then transfer to a sheet of parchment paper set on baking peel or baking sheet.
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12
Holding 1 edge of floured dough in the air with both hands and letting bottom touch parchment, carefully move hands around edge of dough (like turning a steering wheel), allowing weight of dough to stretch it to a round roughly 10 inches in diameter.
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13
Lay round on parchment and adjust shape as needed.
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14
Discard any liquid exuded from tomatoes and toss tomatoes with remaining tablespoon oil and remaining 1/4 teaspoon salt in bowl.
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15
Brush dough round with oil and scatter half of eggplant, half of tomatoes, and half of cheese over it, leaving a 1-inch border around edge.
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16
Line up far edge of peel with far edge of pizza stone and tilt peel down, jerking it gently to start pizza moving.
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17
Slide pizza, along with parchment, onto stone and bake until crust is golden brown and cheese is bubbling, 12 to 15 minutes.
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18
Remove pizza from oven by sliding peel under paper.
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19
Transfer pizza to a cutting board and discard parchment.
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20
While first pizza is baking, assemble second pizza on another sheet of parchment, then bake in same manner.
