Turn on the broiler. Butter both halves of the sandwich buns and stick them under the broiler, sliced-side up, until lightly toasted.

Set a large nonstick sauté pan or cast-iron skillet over medium heat and add olive oil.

Break in the eggs and fry until the whites are set and starting to brown on the edges but the yolks are still runny, about 4 to 5 minutes. (The egg whites will be fully cooked, so no need to flip the eggs over.) Remove to a large plate.

Swab the tops and bottoms of the bun halves with the aioli.

Place a warmed porchetta slice on each bottom half and set a fried egg on top.

Distribute the arugula on top of the eggs, close up the sandwiches, and enjoy!