In a small bowl, whisk together the mayo, mustard, sour cream, and paprika.
Trim the ends off the bread (chef's snack!). Cut the loaf into 3 sections, then slice each section horizontally to make tops and bottoms for 3 sandwiches.
Spread the sauce generously on the inside of each piece of bread, dividing it evenly among the sandwiches.
Layer each bottom half with 2 slices Swiss cheese, 3 slices ham, warm pulled pork, pickle slices, and 2 more slices Swiss cheese. Add the top pieces of bread. Butter the outside top and bottom of each sandwich.
Heat a heavy skillet, cast iron if you have it, over medium heat. Add the sandwiches, then press them down with another heavy skillet or pot. You can also use a sheet pan weighted with a few cans.
Cook for about 3 minutes per side, watching closely and adjusting the heat as needed, until the bread is dark golden and crisp, the cheese is melted, and the filling is warmed through.
Let sandwiches stand for 1 minute, then slice in half and serve immediately.
Notes
Note 1: I either buy this prepared pulled pork or use leftovers from my BBQ Pulled Pork recipe (without the BBQ sauce). Make sure the pork is nice and warm before assembling so it helps heat the sandwich through the middle.Storage: These sandwiches are best fresh, but leftovers can be wrapped in foil and refrigerated for 2–3 days. Reheat in the oven or air fryer until warmed through and crisp.