Bring a pot of water to a boil and have an ice bath ready. Add the broccoli to the pot and cook for 4 to 5 minutes. Broccoli should be bright green and mostly tender. Transfer broccoli to the ice bath to stop the cooking.

Heat oven to 400˚F. Place the slices of bread on a sheet tray and brush with olive oil. Bake for 10 to 15 minutes or so, just until the bread is just starting to toast. Time will depend on how thick your bread slices are.

While the bread is toasting, combine ingredients for the broccoli in a blender or food processor. Puree until the broccoli is in small pieces/the sauce is mostly smooth. Add the broccoli blanching water as needed to thin the puree enough to combine everything. I like the puree to be on the thick/spreadable side.

Assemble the melts by adding a layer of shallots to the toast, followed by the broccoli puree, and cheese. Return the pan to the oven and bake until the cheese has melted and is slightly browning, 10 to 12 minutes.

Remove the melts from the oven and top with fried eggs if desired. Serve the melts with a drizzle of olive oil, red pepper flakes, and fresh parsley.