Heat a 10- or 12-inch nonstick or cast-iron skillet over medium-high heat. Add the vegetable oil and swirl to coat. When shimmering, add the bologna slices in a single layer (they will likely curl slightly). Fry without moving for 1½ to 2 minutes until the bottom edge is deeply browned and crisp. Flip and fry the second side 1 to 1½ minutes until both sides have golden-brown crisp edges. Transfer bologna to a plate and sprinkle evenly with the kosher salt and black pepper.

Assemble sandwich

Spread the mayonnaise on one side of one slice of Texas toast and the yellow mustard on one side of the other slice. On the mayonnaise side, layer 1 slice sharp cheddar and the American cheese. On top of the cheese place all 4 hot crispy bologna slices (you can fold one if needed to fit). Top with the remaining slice of sharp cheddar, then close the sandwich with the mustard-coated bread slice (mustard side inward). Press lightly to compact.

Butter and grill

Reduce heat to medium. Spread 1 tablespoon of butter evenly over the outer face of each slice of the sandwich (the sides that will contact the skillet). Place the sandwich in the preheated skillet, buttered side down. Cook 2½ to 3 minutes until the bottom is deep golden-brown and the cheese begins to melt. Meanwhile add the remaining 1 tablespoon butter to the top slice so it will brown when flipped. Carefully flip the sandwich with a spatula and cook another 2 to 3 minutes until the second side is golden and the cheese is fully melted. If the exterior browns too fast before the cheese melts, lower heat slightly and cover the skillet with a lid for 30–60 seconds to help melt the cheese.

Rest and serve

Transfer the grilled bologna sandwich to a cutting board and let rest 1 minute to allow the cheese to set slightly. Slice in half diagonally (or down the middle) and serve immediately while hot and crisp.