To make the dough, put 2/3 cup warm water in a measuring cup. Add the yeast and sugar and let sit for 5 - 10 minutes, until frothy.
Sift the flour and salt into a large bowl and make a well in the center. Gradually pour in the yeast mixture and 1 tbsp oil. Mix to make a smooth dough. Knead the dough on a lightly floured surface for 10 minutes, until smooth. The dough should be springy and elastic.
Place the dough in a floured bowl, cover and let rise in a warm place for 1-1/2 hours. Preheat the oven to 425°F. Trim the eggplant, then cut into small cubes.
Heat 1 tbsp of the oil in a frying pan and cook the shallots until soft. Add the eggplant, garlic, sun-dried tomatoes, crushed red pepper, thyme and seasoning. Cook for 4 - 5 minutes, stirring frequently, until the eggplant is beginning to soften.
Divide the dough in half and roll out each piece on a lightly floured surface to a 7-inch circle.
Spread the eggplant mixture over half of each round, leaving a 1-inch border, then top with the mozzarella cubes.
Dampen the edges with water, then fold over the other half of dough to enclose the filling. Press the edges firmly together to seal. Place the calzones on two greased baking sheets.
Brush with half the remaining olive oil and make a small hole in the top of each to allow the steam to escape. Bake for 15 - 20 minutes, until golden. Remove from the oven and brush with the remaining oil. Sprinkle with the Parmesan cheese and serve immediately.