Preheat oven with convection to 180°C/360°F. Wash and dry potatoes. With a sharp knife, make thin cuts along the potatoes. Make sure not to cut all the way through the potatoes, so the bottom remains intact.
Melt butter in microwave. Brush half of the butter on the potatoes, sprinkle with sea salt, and place into a baking dish. Bake the potatoes in at 180°C/360°F for approx. 30 min.
Remove from oven. Brush the remaining butter in between the slices on the backs of the potatoes. Return to oven and bake for approx. 30 min. more.
While the potatoes finish baking, peel and roughly chop garlic. Strain tomatoes and chop roughly.
Add garlic and tomatoes to a liquid measuring cup and purée with an immersion blender. Add egg yolks, mustard, and white wine vinegar to the cup, and purée until smooth. Slowly pour in oil while puréeing. Season with sugar and lemon juice to taste. Remove the potatoes from oven. Serve immediately with the aioli dip. Enjoy!