Melt 1 tablespoon vegan butter in a large saute pan over medium heat. Toast the insides of the bread roll until golden brown.
First, spread the yellow mustard onto the bottom half of the bread. Lay down 2 halves of cheese, then the pickle slices, slices of ham, artichoke pulled pork, and then the other 2 halves of cheese. Spread across the garlic aioli onto the inside of the top piece of bread. Then, press the bread on top of the rest of the sandwich. Press down firmly, but gently.
Melt 2 tablespoons of butter in a large saute pan over medium heat.
Once hot cook, place the sandwich into the pan and cook until deep golden brown and crispy, about 2-3 minutes. While cooking, press down on the sandwich reasonably hard, but gently, to ensure the fillings do not squeeze out.
Flip over, turn the heat down to medium-low, and cook until the second side is deep golden brown and crispy and all of the cheese has melted. Continue pressing down on the sandwich. *Cover the sandwich with a lid if the cheese won't melt.