Caramelize the onions
Heat the oil in a skillet over medium heat. Add the sliced onion with a good pinch of salt and 1 tablespoon of the butter. Spread the onions into an even layer in the pan and lower the flame to medium-low. Cook, stirring regularly, until the onion turns soft and develops a rich amber color — this will take about 20–30 minutes. If the pan looks dry or the onions begin to stick, splash in the 1/4 cup water to loosen any browned bits and continue cooking until the onions are deeply caramelized. Keep them warm while you finish the sandwich.

Make the soft-scrambled eggs
Crack the eggs into a bowl, season lightly with salt and pepper, and whisk until blended. Heat an uncoated skillet (no preheating necessary) and pour the eggs straight into the cold pan. Stir gently to break them up, then add 1 tablespoon of butter. Move the pan to medium-low heat and stir slowly and continuously — you want custardy, just-set curds, not dry curds. When the eggs are almost done but still glossy, fold in the chopped chives and remove from the heat; residual warmth will finish them.

Build the sriracha-mayo
In a small bowl, combine the mayonnaise and sriracha in about a 2:1 ratio (2 tablespoons mayo to 1 tablespoon sriracha). Stir until smooth and adjust the heat to your liking.

Toast the brioche and melt the cheese
Spread the remaining butter on the cut faces of the brioche halves and toast them in a skillet over medium heat until golden brown. Place the cheddar slice on one toasted half for a few seconds on the warm pan to let it soften and begin to melt.

Assemble the sandwich
Spread sriracha-mayo on both toasted bun halves. Pile the cheesy, soft-scrambled eggs onto the bottom bun, spoon 1–2 tablespoons of caramelized onion over the eggs, then top with the other bun half. Serve immediately.