Preheat oven to 350 degrees F. Season the pork with salt, pepper and cumin. Heat a few tablespoons of oil in a large Dutch oven until it begins to shimmer. Add the pork and cook until golden brown on all sides.
Add the stock, juices, vinegar, garlic, onions, bay leaves and oregano, bring to a simmer, cover and transfer to the oven. Braise until fork tender (between 180 and 190 degrees F), about 1 1/2 hour to 2 hours. Remove from the oven, transfer pork to a platter and let rest 15 minutes before slicing or shredding.
Strain the braising liquid into a medium pan, bring to a boil and let reduced slightly, adding honey and salt and pepper if needed. Toss some of the braising liquid with the shredded pork to moisten.
Put the sliced cucumber in a medium bowl (which has a fitted lid).
Combine 1 cup of water, the vinegar, dill, sugar, garlic, salt, dill, mustard and coriander seeds in a small saucepan over high heat and bring to a boil. Cook until the sugar and salt is dissolved. Remove from the heat and taste for tartness, add more sugar if too sour and let cool to room temperature.
Pour the cooled liquid over the cucumbers, cover and refrigerate for at least 4 hours and up to 48 hours, stirring the mixture at least once.
Spread both cut sides of the bread with a tablespoon or so of aioli. Add the pork, a few slices ham, and a generous layer of pickles. Top with a few slices cheese.
Press the sandwiches in a hot panini grill until outside is toasty and cheese is melted.
Whisk together mayonnaise, mustard and garlic and season with salt and pepper. Cover and refrigerate for at least 30 minutes before using to allow the flavors to meld.