In a large bowl, combine the flour, salt, sugar, and active dry yeast. Add the lukewarm water and olive oil, and knead until you have a soft, elastic dough. If the dough is too sticky, add a little more flour. If it's too dry, add a little more water.
Cover the dough and let it rise in a warm place for about 1 hour, or until doubled in size.
Meanwhile, prepare the filling: finely chop the garlic and mix it with the pesto. Wash the arugula and let it drain.
Grate the mozzarella cheese and set aside with the other filling ingredients.
Once the dough has risen, turn it out onto a lightly floured surface and divide it into four equal portions. Roll each portion into a circle about 3 mm thick.
Spread the garlic pesto on one half of each circle, then top with arugula and grated mozzarella. Be sure to leave about 1 cm of space around the edges for folding.
Fold the dough in half to form a half-moon shaped calzone. Press the edges firmly to seal, then crimp them with a fork to prevent them from opening 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 before serving fresh, optionally with extra pesto on top or a garlic dip.
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