In a large bowl, combine water, yeast, honey and olive oil. Mix with a spoon, then let sit until foamy, about 10 minutes. Add the flour and salt, stirring with a spoon until the dough comes together but is still sticky. Using your hands, form the dough into a ball and work in the additional 1/2 cup flour, kneading it on a floured surface for a few minutes. Rub the same bowl with olive oil then place the dough inside, turning to coat. Cover with a towel and place in a warm place to rise for about 1 to 1.5 hours.
After the dough has risen, punch it down and place it back on the floured surface. Divide it in half and use only one half for this pizza recipe. Save the other half - stick it in the fridge for a few days, or even freeze it!
Using a rolling pin or your hands, form it loosely into your desired shape. Sprinkle a bit of flour or cornmeal on a 10x15 inch baking sheet. Place the dough on top. You can also use a pizza stone or cook this in your pizza oven if you wish!
Preheat the oven to 450 degrees F.
Make the vodka sauce. To make a starchy pasta water without cooking pasta, combine 1 cup of water with ½ teaspoon of cornstarch and ¼ teaspoon salt. Bring it to a boil in a saucepan, then turn off the heat!
Heat the butter in a saucepan over medium heat. Once melted, stir in the shallots and garlic with a big pinch of salt and pepper. Cook, stirring often, until the onions soften. Add in the tomato paste and stir, cooking for 5 minutes. The color will darken - this is where the flavor is!
Stream in the vodka and stir, scraping the brown bits and flavor from the bottom. Stir in ½ cup of the reserved (or makeshift) pasta water. Stir in the heavy cream and parmesan, stirring until the cheese melts. Taste and season the sauce with more salt and pepper if needed. This will make more sauce than needed. Store it in the fridge for a few days or use it to make another pizza!
To make the pizza, spread about ½ to ¾ cup of vodka sauce all over the pizza dough. Top with the fresh mozzarella. Sprinkle on some parmesan cheese.
Place the pizza on the sheet in the oven for 15 to 20 minutes. Check on it after 15 minutes - if it’s not deeply golden and melty yet, give it another 5 minutes. Once golden and bubbly, remove it and let cool before slicing. Sprinkle with basil and extra parmesan and serve!