In a medium pot, combine the potatoes and enough salted water to cover by 1 inch. Bring to a boil, then lower the heat and simmer until just tender, about 12 minutes; drain.
Meanwhile, in a shallow bowl, beat the eggs. In another shallow bowl, place the breadcrumbs. Season the chicken with salt and pepper and, working with 1 piece at a time, coat with the breadcrumbs, eggs and breadcrumbs again. Transfer to a sheet of wax paper.
In a large nonstick skillet, heat 2 1/2 tablespoons olive oil over medium-high heat. Add the chicken and cook, turning once, until golden-brown and cooked through, about 8 minutes. Transfer to a cutting board and let rest for 5 minutes before slicing. Wipe out the skillet.
In the same skillet, heat the remaining 1 1/2 tablespoons olive oil. Add the coleslaw mix and bell pepper; season with salt and pepper. Cook, stirring often, until wilted and tender, about 5 minutes. Remove from the heat and stir in the scallions. Coarsely mash in the potatoes and cheese; season with salt and pepper. Serve with the chicken.