Pierce the potatoes with a fork a few times on all sides. Place on a baking sheet and bake until fork-tender, 1 hour to 1 hour 20 minutes. Let cool slightly. (Leave the oven on.)
Meanwhile, melt 4 tablespoons butter in a small skillet over medium heat. Add the shallot and cook until softened but not browned, about 5 minutes. Transfer to a small bowl. Melt the remaining 2 tablespoons butter in a small bowl in the microwave.
Once the potatoes are cool enough to handle, use a paring knife to make a slit lengthwise down the top of each, cutting about halfway through. Make another small slit across the first to make a "t". Gently push the sides of the potato with your hands to loosen the filling. Scoop out the flesh to a large bowl with a spoon or melon baller, leaving a 1/4 inch of flesh in the skins to preserve the shape of the potatoes. Return the potato skins to the baking sheet.
Mash the potato flesh until smooth. Add the shallot butter mixture, creme fraiche, chives and lemon zest, then season with salt and pepper. Gently fold in the crabmeat.
Spoon the mixture evenly into the potato skins. Drizzle with the remaining melted butter. Bake until browned, 20 to 25 minutes. Top each potato with a dollop of creme fraiche, a sprinkle of chives and a spoonful of caviar. Serve immediately.