In a large bowl, stir together honey, yeast, and 1⁄4 cup water, heated to 115°; let sit until foamy, about 10 minutes. Stir in beer and oil until smooth. Add flour and salt; stir with a wooden spoon until dough forms. Transfer to a lightly floured work surface and knead until smooth, about 8 minutes. Cover with plastic wrap and let sit in a warm spot until doubled in size, about 1 1⁄2–2 hours.
Meanwhile, make the onion puree: Heat 2 tbsp. oil, thyme, white onions, bay leaf, and salt and pepper in a 12″ skillet over medium-low heat, and cook, stirring occasionally, until onions are very soft but not browned, about 30 minutes. Remove and discard thyme stems and bay leaf. Transfer onions to a food processor or blender and puree until smooth; set aside.
Make the onion compote: Heat 1 tbsp. oil in a 12" skillet over medium heat; add leeks, season with salt and pepper, and cook, stirring occasionally, until very soft but not browned, about 15 minutes. Transfer to a bowl and set aside. Heat remaining oil in skillet, add shallots and red onions, and season with salt and pepper; cook, stirring occasionally, until very tender and lightly browned, about 18 minutes. Transfer to bowl with leeks, and stir to combine; set aside.
Uncover dough and cut into quarters; shape each quarter into a smooth ball. Lightly flour dough balls and transfer to a floured 9" × 13" baking pan; cover with plastic wrap. Let sit in a warm spot until doubled in size, about 1 1⁄2–2 hours. Heat oven to 500°.
Place one piece dough on a lightly floured work surface and flatten with your fingertips. Pick up dough circle and gently feed edges of dough between your thumbs and forefingers, letting the weight of the dough stretch edges until the circle of dough is 12″ in diameter. Place dough circle on a parchment paper-lined baking sheet, and working quickly, spread about 2 tbsp. onion puree over dough, leaving a 3⁄4″ border around edge; sprinkle evenly with about 1⁄4 cup onion compote. Sprinkle one-quarter of the pecorino over onions, and transfer to oven. Bake until browned and crisp at the edges, about 12 minutes. Repeat with remaining dough balls, puree, compote, and pecorino. Sprinkle each pizza with one-quarter each of the scallions and chives before serving.