Lay potatoes out in a single layer on a sheet pan. Drizzle with olive oil and season with salt and pepper. Toss the potatoes around so they are evenly coated and pop in the oven to roast for 35 to 40 minutes until tender.
Cut a heads of garlic through the middle, horizontally, and put in a foil pouch with olive oil, thyme, water, and salt and pepper. Seal the pouch around the edges and roast in the oven with the potatoes - this will be used for the aioli.
When the potatoes are done and warm enough to handle make an X cut on 1 side of the potato and then squeeze gently from the bottom to form a flower-like shape. Heat a pot of oil for deep frying to 350 degrees F and fry the potatoes until crispy and golden. Drain on paper towels and season with kosher salt.
Remove the garlic from the pouch and squeeze out the flesh. Add the roasted garlic, egg, lemon juice, salt and pepper in a blender and process. Pour in the olive oil in a slow, steady stream until the aioli emulsifies. Fold in sour cream and adjust seasoning with salt and pepper. Garnish with chives.