Set a large skillet over medium-high heat with the olive oil. Once hot, add the ground sirloin and begin breaking it up with a wooden spoon. Cook until crispy and no longer pink, about 10 minutes. Add the onion, bell pepper and garlic and cook until softened, about 5 minutes.
Add the ketchup, tomato paste, Worcestershire sauce, mustard, salt, pepper, cayenne, and chicken stock. Bring to a simmer and cook until the sauce has thickened and almost evaporated completely, about 2-3 minutes. Remove from the heat and let cool.
To assemble the sandwiches, lay out 4 bread slices in a single layer. Brush the bread with 3 tablespoons of the softened butter, then turn the bread over. Top each bread slice with 1/4 cup shredded cheese. Spoon the Sloppy Joe mixture on top of the cheese. Top with remaining cheese and top each with a slice of bread. Brush with remaining butter.
Heat a skillet or griddle over moderate heat. Working in batches, cook sandwiches until bread is crisp and golden and the cheese has melted, about 3 minutes per side. Transfer to a cutting board and cook the remaining sandwiches.
To serve, cut each sandwich in half and skewer with a few pickle slices on top.