In a blender, combine the mayonnaise, ½ cup orange juice, 1 tablespoon lime juice, garlic, cumin, and salt until smooth. Add the cilantro and blend until smooth and creamy, about 30 seconds.
The order of assembly is important here! Spread the bottom slice of bread with mustard and sprinkle with the Gouda. Build the rest of the sandwich on the top slice of bread, starting with the Gruyère, followed by a slice of ham and the Barbacoa. Place a liberal layer of pickles over the pork, and top that with a full-on slathering of the cilantro mayonnaise. Flip the bottom up onto the top and press down firmly to compact.
Heat a panini press to medium and coat lightly with nonstick spray or a brush of grapeseed or canola oil. You really want a nice, slow melt and toast. Place the sandwiches on the press, top side down, and close the top. Cook for 3 minutes, or until the bread is toasty and golden brown and the cheese starts melting.
Alternatively, if you’re using a griddle, heat to medium and coat the pan lightly with nonstick spray or a brush of oil. You do not want a smoking-hot griddle here, or it will char the bread. Place the sandwiches on the griddle, top side down, and place a baking sheet on top of the sandwiches and then place some heavy weights on top to compress the sandwiches. Cook until the bottom surface of the bread is golden brown and you start to see the cheese melting, 3 minutes. Remove the baking sheet from the sandwiches, brush the top of the bread lightly with oil, then flip. Replace the baking sheet and weights on top and continue cooking until the sandwiches are golden brown and crispy and you can see the rest of the cheese melting, another 2 to 3 minutes. Remove the sandwiches, flip over, and cut in half on the diagonal. Serve the sandwiches with the remaining cilantro mayonnaise for dipping.