Oil griddle top and heat to medium-low.
Ladle approximately eight-ounces scrambled egg mix onto the griddle in a single thin sheet the same width as the sliced cheese.
Toast two slices of brioche on the oiled griddle, next to the egg sheet.
Season eggs with salt, and sprinkle with 1 tbsp. finely sliced chives.
As eggs begin to set, place a slice of cheddar and Swiss side by side in the center of the sheet of eggs.
Fold the top and bottom edges of the egg sheet over the sliced cheese.
Place slice of Jack cheese on top of the folded eggs that are now covering the cheddar slice.
Fold the egg sheet in half to cover the Jack cheese, and top with finely grated pecorino cheese.
To finish, spread ½ tbsp. of avocado chive butter on each piece of toasted brioche. Then add the folded eggs and cheese to the sandwich, slice diagonally with serrated knife. Garnish with more finely slice chives and a pinch of sea salt.
Purée chives and cold water in a blender until very smooth and well combined.
Add butter, chive purée and avocado to the bowl of a food processor, and purée until smooth.*
Add salt and lemon juice and adjust salt to taste.
*Note: Be careful not to purée past smooth, as the heat from the processor will break the butter.