In a large skillet, heat the oil over medium-high. Add the jackfruit to the pan and sauté until soft, about 5 minutes.
Reduce the heat to medium-low, season the jackfruit with salt, cumin, and dried oregano, and add BBQ sauce and lime juice. Stir together and let them simmer, stirring often, 5 to 7 minutes. Remove pan from heat and use a fork to flake jackfruit into small pieces. Set aside.
On each slice of bread, add a spread of mustard. Then layer one slice of bread with swiss cheese, jackfruit, ham and pickles. Close sandwiches and add a light coat of oil or vegan butter to the outside of the bread.
Grill the sandwich on a panini press or in a skillet, for about 5 minutes, or until outside of the bun is golden brown, and cheese is melted, flipping once. Slice and serve with chips.
If making sandwiches ahead, wrap them in foil and store them in the refrigerator. When ready to enjoy, reheat them with the same steps as above.