Preheat your oven to 400°F (200°C). Slice the top quarter-inch off the head of the garlic to expose the raw cloves inside. Place the garlic on a square of aluminum foil, drizzle with 1 tsp of olive oil, and wrap it tightly. Roast for 40-45 minutes until deeply golden and blistered. Let cool, squeeze out the cloves, and mash into a paste.

In a tall, narrow jar, add the room-temperature egg, fresh lemon juice, Dijon mustard, the roasted garlic paste, kosher salt, and coarse black pepper. Pour the 1 cup of neutral oil directly on top of the wet ingredients. Let the mixture sit for 30 seconds.

Lower your immersion blender to the bottom of the jar. Turn it on high speed and hold it perfectly still for 10-15 seconds until a pale yellow emulsion forms. Slowly tilt and pull the blender upward to incorporate the remaining oil.

Cook the french fries until crispy golden brown and transfer them to a black cast iron skillet. Drop a large dollop of the aioli in the center. Sprinkle finely chopped fresh parsley and coarse black pepper exclusively over the aioli. Serve immediately.