Pound pork and chicken cutlets to 1/8 to 1/4 inch thick and season with salt and pepper. Make marinade by whisking together olive oil, lemon and lime juices, vinegar and 1 Tbsp. Cuban Island Spice. Place chicken in one quart-size zip top bag and pork in another. Divide marinade between the bags, seal and refrigerate for 30 min.
Mix mayonnaise, mustard and capers together in a small bowl and set aside.
Transfer marinated chicken to one plate and pork to another and discard marinade. Sprinkle cutlets generously with remaining 2 Tbsp. Cuban Island Spice. Heat oil to medium-high in a large skillet. Pan fry cutlets in batches until cooked through, 2 to 3 min. per side. As you flip each cutlet over, top with 1 slice of cheese. Transfer cooked cutlets to clean plates (keeping chicken and pork separate) and tent with foil.
Slice cutlets into ½-inch thick strips. Spoon mayo-mustard mixture on bottom half of hoagie, line with alternating slices of chicken and pork strips and top with pickle slices.
Cover with top half of hoagie and place in clean skillet over medium-high heat. Use another heavy pan or lid to press sandwich down as it cooks until bread is toasted and sandwich is heated through, 2 to 3 min. per side. Serve warm.