Preheat the oven to 450 degrees . Line 2 baking sheets with parchment paper and fill a large bowl with cold water. Using a mandoline or a food processor fitted with a slicing disk, thinly slice the potatoes and place in the water. Change the water twice; drain. Arrange the potato slices in a single layer on paper towels and pat dry.
In a large bowl, toss the potato slices with 1/4 cup olive oil to coat. Arrange half the slices in an even layer on the prepared baking sheets and season with salt. Bake until golden brown, about 25 minutes. Transfer to paper towels and let cool. Repeat with the remaining slices.
Meanwhile, in a medium saucepan, cook the bacon over medium heat, stirring occasionally, until crisp. Using a slotted spoon, transfer the bacon to a small bowl and set aside. Discard the bacon fat. Add the remaining 2 tablespoons olive oil, the carrot, onion and garlic to the saucepan and cook over medium heat, stirring, until tender, 6 to 8 minutes. Add the beans, cover and cook over medium-low heat for 5 minutes. Let cool.
Using a food processor, puree the bean mixture, yogurt and 2 or 3 tablespoons water until nearly smooth. Add the reserved cooked bacon and pulse to incorporate. Season to taste with salt and pepper. Serve the dip with the potato chips.