Remove the dough from the refrigerator to bring to room temperature. Place an oven rack in the center of the oven; preheat to 475°F. Wash and dry the fresh produce. Thinly slice the mushrooms. Halve, peel, and thinly slice the shallot. Peel and roughly chop 2 cloves of garlic. Thinly slice the chives. In a bowl, combine the béchamel and mozzarella (tearing into small pieces before adding); season with salt and pepper.

In a medium pan (nonstick, if you have one), heat a drizzle of olive oil on medium-high until hot. Add the sliced mushrooms in an even layer. Cook, without stirring, 3 to 4 minutes, or until lightly browned. Add the sliced shallot. Cook, stirring occasionally, 2 to 3 minutes, or until slightly softened. Add the chopped garlic; season with salt and pepper. Cook, stirring occasionally, 1 to 2 minutes, or until softened. Turn off the heat.

Lightly oil a sheet pan. Using your hands, gently stretch the dough to about 1/4-inch thickness. Carefully center the dough on the sheet pan. Shape towards the edges of the pan, maintaining an even thickness (if the dough is resistant, let rest 5 minutes). Using a fork, gently prick the surface of the dough all over to prevent air bubbles from forming. Leaving a 1-inch border around the edges, evenly top the dough with the béchamel-cheese mixture, cooked vegetables, and a drizzle of olive oil; season with salt and pepper. Bake, rotating the sheet pan halfway through, 17 to 22 minutes, or until the edges are lightly browned and the cheese is melted. Remove from the oven and let stand at least 2 minutes.

Meanwhile, in a bowl, combine the honey, 1/4 teaspoon of warm water, and as much of the red pepper flakes as you'd like, depending on how spicy you'd like the dish to be. Carefully transfer the baked pizza to a cutting board. Evenly top with the hot honey and sliced chives. Cut the finished pizza into equal-sized pieces. Enjoy!