Place the potatoes in a large saucepan of salted water, bring the water to a boil over medium-high heat and boil for about 10 minutes, or until the potatoes are just tender; a sharp knife should pierce a piece of potato easily and the potato should slip off the knife without falling apart.
Meanwhile, in a large, heavy, nonstick skillet, cook the bacon over medium-high heat for about 6 minutes, or until crisp and golden brown. Transfer the bacon to a plate lined with paper towels to drain.
In a small saucepan, bring the broth to a simmer over high heat. Add the shallots, cornichons, and capers, reduce the heat to medium-low and simmer very gently for 2-3 minutes, or until the shallots soften slightly. Remove from the heat.
Drain the potatoes in a colander and gently shake them to release excess moisture. Transfer the potatoes to a large bowl.
Add one-third of the broth mixture to the hot potatoes and, using a silicone spatula, gently fold and turn the potatoes in the hot broth for about 2 minutes, or until most of it has been absorbed. Repeat 2 more times, adding all of the solids and just enough of the broth to moisten. The potatoes should break down a bit. Set aside.
Gently fold the bacon and parsley into the warm potatoes then gently fold in the aioli. Season to taste with salt and pepper. Serve warm, at room temperature or chilled.
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