To char the scallions: Position a rack roughly 5 to 6 inches away from the broiler and turn your broiler on. Trim the ends from the scallions, separate the whites from the greens, and place both on a sheet tray. Drizzle with olive oil and kosher salt, tossing to coat well. Place the tray under the broiler and roast until the whites are quite tender and the greens have blackened. If you find the greens are done well before the whites, use tongs to remove them and set them aside. Cook until you have plenty of crispy bits. This takes around 6 minutes under my broiler, but your oven makes a huge difference on time, so check often! Once the scallions are to your liking, remove them from the oven and let them cool.
To make the cream cheese spread: Place the room temperature cream cheese in a food processor along with the whites and greens of the charred scallions and 2 tablespoons of heavy cream. Run until well combined and the cream cheese is lighter in texture. If the cream cheese still feels dense, add another tablespoon or two of the heavy cream and continue. Taste and add more salt if desired.
To assemble the sandwiches: Spread a tablespoon or so of cream cheese on one piece of bread, layer with lettuce and cucumbers, and then spread another tablespoon of cream cheese on the top piece of bread before smooshing together.
