Heat a griddle pan over medium-high heat. Lightly brush the sourdough slices with olive oil and place on one side of the pan.
Brush the heads of the mushrooms with olive oil and place head side down on the griddle pan.
When the mushrooms are smoky and have griddle marks, remove them from the pan and allow to cool.
Check the bread. When the pieces have good griddle marks, brush the tops of the pieces with olive oil and turn the pieces over.
When the mushrooms have cooled a little, slice them into 2 or 3 pieces.
Warm a saucepan over medium heat, add a little olive oil and butter and add the shallots and cook for a minute before adding the garlic and the mushroom slices. Cook until the pieces of mushroom are lightly coloured and remove from the heat. Add a sprinkle of chives and salt. If you're not doing the eggs immediately, transfer the mushrooms to a sheet pan and keep warm in a low oven.
Remove the toasted sourdough pieces, slice in half and place on warmed plates for serving.
Gently scramble the eggs in a little butter and olive oil until they are glossy but not fully cooked (you can cook them through but they will keep cooking off the heat). Mix through some chives and distribute between the serving plates on top of the toasted sourdough slices.
Using a spoon, top the scrambled eggs with the mushroom mixture and top with the remaining chives.
Serve immediately.