Heat a frying pan with butter. Fry onions and tomato in oil over medium heat for 5 minutes. To help soften the onions, add a splash of water and continue to cook another few minutes.
Combine miso, soy, honey along with another small splash of water and pour over onions. Cook until sauce is thick and coating the onions and tomatoes nicely. Remove from the pan and set aside.
Place brioche bun halves under the grill and brown slightly. Butter each side, then on the top half bun cover with slices of cheddar cheese. Return to the grill until the cheese has melted.
In a clean frying pan, heat some oil and fry egg, sunny side up, until crispy. Season with salt.
Assemble breakfast burger: On the bottom bun layer lettuce, miso tomato onions, fried egg and finish with cheesy brioche bun top.