Heat 1 1/2 teaspoons of the olive oil in a large skillet over medium-high heat. Add the beef and capicola, and cook, stirring, until just beginning to crisp on the edges, about 3 minutes; transfer to a plate with a slotted spoon. Add the remaining 1 1/2 teaspoons olive oil, onions and bell peppers and cook, stirring, until crisp-tender, about 5 minutes.
Scoot the vegetables to the side of the pan. Add the tomato paste to the open space and toast, stirring for a minute, and then combine with the vegetables. Add the cherry pepper brine and 1/2 cup water, bring to a simmer and return the meat to the pan. Simmer just until the sauce coats the meat and vegetables, about 1 minute.
Divide the meat and vegetables among the rolls. Top with the cherry peppers and provolone. Bake on a baking sheet until the cheese is melted and bubbly, about 2 minutes.