Melt one third of the butter in a large non-stick skillet over medium heat. Add hot dog slices and cook, turning occasionally, until browned on all sides. Transfer to a plate. Add both bread slices and cook, swirling occasionally, until lightly golden brown on bottom, about 2 minutes.
Transfer bread to a cutting board, toasted-side-up. Top each with a slice of cheese. Add the grilled hot dogs to one slice of bread and pile chili on top in the center of the hot dogs. Close sandwich with the untoasted sides of bread facing out.
Melt another piece of butter in the skillet and reduce heat to medium low. Add sandwich and cook, swirling occasionally, until golden brown, about 5 minutes. Carefully flip sandwich, melt the remaining butter in the skillet, and cook sandwich, swirling occasionally, until second side is golden brown and cheese is melted, about 5 minutes.
Recipe adapted from www.seriouseats.com