Preheat the oven to 425°F
Pierce the potatoes all over with a fork. Microwave on high for 4 minutes, then transfer to a large rimmed baking sheet and bake until soft inside, about 45 minutes
While the potatoes are baking, combine the beans, aioli, garlic, shallot, olive oil, chives, lemon juice, mustard, salt, pepper, and vinegar in the bowl of a food processor or blender and puree until very smooth, scraping down the sides of the bowl as needed. Transfer to a large bowl
When the potatoes are ready, let them cool for a few minutes until they're easy to handle and increase the oven temperature to 450°F
Slash open the potatoes and use a fork to flake the flesh into the bowl with the bean mixture. Gently mix in the potatoes until incorporated
Let the mixture stand for 20 minutes so the potatoes can absorb the moisture
Transfer the mixture to a 3- to 4-cup gratin dish or small casserole. Bake until the top is bubbly and lightly browned, 15 to 20 minutes
Serve with crusty bread