Preheat your oven to 400°F (200°C). Bake the eggplant for about 30 minutes, or until it's completely soft and the skin is dark.
Let the baked eggplant cool slightly, then scoop out the flesh and place it in a bowl.
Add the minced garlic, lemon juice, olive oil, and oregano. Use an immersion blender to purée until smooth.
In a large bowl, combine the flour, salt, sugar, and dry yeast. Add the lukewarm water and olive oil. Knead until you have a soft, elastic dough.
Cover the dough and let it rise in a warm place for about 1 hour, or until it has doubled in size.
Turn the risen dough out onto a lightly floured surface. Divide it into four equal portions. Roll each portion into a circle about 3 mm thick.
Spread eggplant dip on one half of each circle, then sprinkle with grated mozzarella. Leave about 1 cm of space around the edges for folding.
Fold the dough over to create a half-moon shaped calzone. Press the edges together firmly to seal, then use a fork to crimp the edges, ensuring they don't open during baking.
Brush the top of each calzone with a beaten egg to give it a golden-brown color after baking.
Bake in a preheated oven at 400°F (200°C) for about 20 minutes, or until the calzones are golden brown.
Let cool slightly, then serve fresh, optionally with extra eggplant dip on top.
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