Prepare the cooked components of your crudites platter: Place the eggs in a medium saucepan and cover with cold water. Bring to a boil, simmer for 3 minutes, and drain. Allow to cook until room temperature in the shells. Meanwhile, replace the water in the saucepan and add the shrimp. Bring to a simmer (along with a splash of lemon juice) and cook until pink and curled, about 2 minutes, depending on the size. Drain and immediately rinse under cold water to stop the cooking. Finally, add the baby potatoes to the saucepan and cover by 1 inch with cold, salted water. Bring to a boil and cook until fork tender, about 10 minutes. Drain, rinse with cool water, and cut in half.

Meanwhile, grill the scallions: heat an indoor grill pan or outdoor grill until smoking. Brush the scallions with olive oil and grill until charred on both sides, 2 minutes on each. Set aside to cool completely.

Last, make the aioli: In a small mixing bowl or bowl of a food processor, whisk or pulse the egg yolks, mustard, lemon juice, and garlic until smooth. Working slowly, add 1 teaspoon of the olive oil and whisk or pulse until incorporated. Repeat with 3 additional teaspoons. Once the oil is emulsifying easily, slowly drizzle in the remaining olive oil, followed by the coconut oil, whisking or pulsing throughout until thick. Add the salt and cooked scallions. Puree until smooth and adjust seasoning as necessary.