Melt 2 tbsp. butter in a large nonstick frying pan over medium-high heat. Add onion, reduce heat to medium-low, and cook, stirring often, until golden brown, about 20 minutes. Scrape into a bowl and set onion and pan aside.
Meanwhile, in a small bowl, whisk Sriracha and mayonnaise to blend. Set aside, chilled.
For brioche buns, preheat oven to 200°. Lay buns cut-side down on a rimmed baking sheet and loosely drape with foil and set in oven to warm (about 20 minutes). For brioche bread, preheat broiler with rack set 5 in. from heat. Cut 1-in. slices of bread and set on a rimmed baking sheet. Broil, turning once, until lightly toasted, about 30 seconds per side.
In a medium bowl, stir eggs with salt until blended but not frothy. Over medium-high heat, melt remaining 4 tbsp. butter in pan used for onion. Pour in eggs and cook, stirring gently, until eggs are halfway set, 2 to 3 minutes. Remove from heat, add chives, and stir occasionally until eggs are softly set, 1 to 3 minutes more. Transfer eggs to a medium bowl and wipe pan clean with paper towels.
Heat pan over medium heat. Meanwhile, set buns on plates and smear Sriracha mayo on cut sides. Pile eggs on bun bottoms. Set cheese in warm pan and cook just until softened, 30 to 45 seconds. Using a wide spatula, set cheese on eggs. Sprinkle with onions and cap with bun tops.