Place racks in upper and lower thirds of oven; preheat to 350°. Grease a 13×9″ baking dish with unsalted butter. Line with parchment paper, leaving overhang on long sides; grease parchment with unsalted butter. Place 2 medium red onions, thinly sliced, on a rimmed baking sheet. Drizzle with 2 Tbsp. extra-virgin olive oil, sprinkle ½ tsp. Diamond Crystal or ¼ tsp. Morton kosher salt over, and toss to coat. Arrange in a single layer.
Whisk 12 large eggs, 1 jalapeño, finely chopped, ½ cup heavy cream, and remaining 1½ tsp. Diamond Crystal or 1 tsp. Morton kosher salt in a small bowl to combine; season with freshly ground pepper. Transfer to prepared baking dish.
Place pan with egg mixture on upper rack and baking sheet with onions on lower rack in oven and bake until egg mixture is puffed around the edges and mostly set in the center (it should only jiggle slightly when tapped), 20–25 minutes. Remove egg mixture from oven and let cool. Transfer onions to upper rack and continue to bake until softened and lightly browned around the edges, 10–15 minutes more. Let cool.
While the onions are cooling, heat broiler. Butter cut sides of 8–12 English muffins, split; broil until toasted and golden, about 3 minutes. Let cool. Decrease (or increase) oven temperature to 375°.
Mix ½ cup mayonnaise and 2 Tbsp. Calabrian chile paste or 1–2 Tbsp. sriracha in a small bowl; set aside.
Run an offset spatula around edges of pan to release egg sheet, then remove from pan. Cut egg evenly into 8–12 pieces, depending on the size of your English muffins. To build sandwiches, stack an egg piece, 1 slice American cheese or cheddar, and 1–2 Tbsp. onions on bottom half of each English muffin; close up. Wrap individually in parchment paper.
If not making ahead, place sandwiches on a clean baking sheet; bake until heated through and cheese is melted, 9–12 minutes. (You can also microwave 30–45 seconds or leave unwrapped and heat in a cast-iron skillet over medium, about 2 minutes per side.) Unwrap and spread 1 Tbsp. chile mayo over cut side of top half of each English muffin just before serving.
Do ahead: Sandwiches (without chile mayo) can be made 3 months ahead. Freeze in parchment paper. Bake in a 375° oven, 10–15 minutes, microwave on high in 30-second bursts, 30–60 seconds, or unwrap and heat in a cast-iron skillet over medium, about 3 minutes per side, to warm. Make chile mayo and spread over sandwiches just before serving.