Eggplant Mixture: Combine oil and garlic in small bowl. Let stand 30 minutes. Place eggplant in colander. Sprinkle with salt and let stand 30 minutes. Drain eggplant and pat dry. Heat half of garlic-oil mixture in heavy large skillet over medium heat. Add eggplant and saute until tender, about 8 minutes. Increase heat to high. Add vinegar and cook until almost no liquid remains in skillet, about 1 minute. Season to taste with pepper. Transfer to large bowl. Heat 1 teaspoon garlic-oil mixture in heavy small skillet over medium heat. Add pine nuts and saute until golden, about 2 minutes. Add to eggplant. (Can be prepared 1 day ahead. Cover and refrigerate remaining garlic-oil mixture and eggplant mixture separately. Bring to room temperature before using. )
Dough: Place 1/4 cup warm water in bowl of heavy-duty mixer; sprinkle yeast over and stir to dissolve. Let stand 10 minutes. Add remaining 3/4 cup water, oil, salt and then 3 cups flour; stir to combine. Attach dough hook to mixer and beat until dough pulls away from sides of bowl, about 2 minutes. Turn out dough onto lightly floured surface and knead until smooth and elastic, adding more flour if sticky, about 10 minutes. Lightly oil large bowl. Add dough, turning to coat. Cover with plastic wrap and let rise in warm draft-free area until doubled in volume, about 1 hour. While dough rises, begin preparation for assembly and baking. Assembly And Baking: Position rack in lowest third of oven. Place baking stone*, baking tiles * or heavy large rimless baking sheet on rack in oven. Preheat oven to 450 degrees for 30 minutes. Mix cheeses together in bowl.
Punch dough down. Divide into 4 pieces. Roll out 1 dough piece on lightly floured surface to 8-inch round. Brush dough with some of garlic-oil mixture, leaving 1-inch border. Spread 1/4 of cheese mixture over half of dough, leaving 1-inch border. Cover cheese with 1/4 of eggplant mixture, 1/4 of bell pepper and 1/4 of sliced basil leaves. Brush edges of dough with water. Fold dough in half, covering filling but allowing bottom edge of dough to show. Fold bottom edge over top edge and crimp to seal. Repeat with remaining dough, garlic-oil mixture, cheese mixture, eggplant mixture, bell pepper and sliced basil leaves, forming total of 4 calzones. Cover calzones and let stand 15 minutes.
Using large metal spatula, transfer calzones to stone or preheated baking sheet in oven. Bake until golden brown and crisp, about 12 minutes. Transfer calzones to platter. Garnish with basil sprigs and serve.