Make the mojo sauce by heating the oil and garlic and cooking until just lightly colored, just a minute or two. You don't want burned garlic. Stir it often and watch it sizzle a bit but don't let it darken too much.
Stir in the rest of the ingredients, then put the pan back on the heat for another minute, stirring. Cool the sauce. You can keep it covered in the 'frig for a week.
Mix the mojo sauce into some mayonnaise, about 1 Tbs mayo and 2 Tbs mojo sauce per sandwich.
Spread the insides of the rolls with mojo sauce, then layer the meats, pickles, onions, tomatoes, and cheese. Put the top on and spread a bit of butter on both sides of the roll.
Grill in a sandwich press or pan and cook over low heat, about 5 minutes per side, pressing down with some weight. I used a big cast iron skillet. You want the meat warmed and the cheese starting to melt.