Special equipment: a stocking-shaped cookie cutter; 2 circular cookie cutters, one small and one larger
For the pesto: In the bowl of a food processor, combine the basil, lemon juice, pine nuts and garlic, and pulse until coarsely chopped. Add the oil and process until combined, and then add the cheese and some salt and pepper, and pulse until it all comes together. Use immediately or store in airtight container until ready to use.
For the dough: In a medium bowl, whisk together 1 cup warm water, the sugar and yeast until frothy. Add about half of the flour and mix with a spoon until it forms a loose dough, then add the oil and salt and continue mixing until combined. Transfer the dough to the bowl of a stand mixer fitted with the dough hook and mix on low speed, adding the remaining flour a little at a time. Knead the dough on medium speed until it springs back when poked, 3 to 5 minutes. Place the dough in an oiled bowl to rest for 5 minutes.
For the pizzas: Preheat the oven to 450 degrees F. Line two baking sheets with parchment paper. Separate the pizza dough into two equal parts and roll each part out to 1/4-inch thick on a floured surface.
Using a stocking-shaped cookie cutter, cut out the pizzas from one half of the dough, then place the pizzas on a prepared baking sheet. Spread some pesto over the dough. Place the rectangles of mozzarella at top of the stocking. Sprinkle the basil over the pesto, and bake for 12 to 15 minutes.
Using one large circular cookie cutter and one smaller circular cookie cutter, cut out equal amounts of each size from the other half of the dough. Overlap the smaller circle on the large circle to create a snowman shape and place on a prepared baking sheet. Drizzle the snowmen with a little olive oil and then place the garlic over the top if using. Spread the ricotta to cover the pizzas and sprinkle the grated mozzarella over the top. Place black olive slices for buttons and eyes, and the orange bell peppers for the nose. Bake for 12 to 15 minutes. Serve immediately.
If necessary, make the dough ahead of time and hold overnight in the refrigerator. Rest the dough at room temperature for 1 hour before rolling out. To serve at a party, prep the pizzas ahead of time, then cook about 10 minutes before the guests arrive so they are hot and fresh when the party starts.