In the bowl of a stand mixer, sprinkle the yeast and sugar over the warm water. Whisk to combine and let sit until bubbles form on the surface, about 5 minutes. Add the flour, salt, and 1 tablespoon of olive oil and mix with the dough hook on low speed until dough comes together, about 2 minutes. Scrape the bowl, then increase speed to medium-low and mix until well incorporated and the dough is sticky, 2 to 3 minutes more.

Drizzle the top of the dough and sides of the bowl with the remaining olive oil. Flip the dough to coat in the oil. Cover the bowl with plastic wrap or a clean kitchen towel and let it rise until doubled, about 1 hour 30 minutes.

About 30 minutes before you're ready to make the pizzas, place a pizza stone, steel, or an upside-down baking sheet on the upper third rack of the oven and preheat to 500°F. Cover an upside-down baking sheet with a 14-inch piece of heavy-duty foil. Lightly brush the foil with olive oil and dust with flour.

On a lightly floured surface, cut the dough in half and shape into tight balls. Place the balls about 3 inches apart on a floured surface, cover, and let rest until the dough springs back slowly when pressed, about 10 minutes.

On a lightly floured surface, pull and stretch one of the dough balls into a 10-inch round and transfer it to the prepared baking sheet.

Brush the crust with olive oil. Sprinkle with half the parmesan, garlic, shallots, low-moisture mozzarella, and fresh mozzarella. Top with half the lemon slices and brush with olive oil.

Using the upside-down baking sheet as a pizza peel, slide the pizza with the foil onto the preheated baking stone. Bake until the crust is browned, cheese is bubbling, and edges of lemon are caramelized, 12 to 16 minutes. Drizzle with half the honey and sprinkle with half the basil just before serving. Repeat with the remaining dough and ingredients.