Special equipment: a smoker (the stand-alone or grill-top variety) and 2 cups pecan wood chips soaked in water for 20 minutes
For the pork tenderloin: Combine the paprika, salt, garlic powder, black pepper and cayenne in a small bowl. Store in a tightly sealed container for up to 6 months.
Clean the pork tenderloin of any silverskin or tough membranes with a sharp paring knife. Rub the tenderloin well with 2 tablespoons of the seasoning mix. Let sit for 1 hour at room temperature or up to overnight in the fridge.
Set up a smoker according to the manufacturer's instructions with 2 cups soaked pecan chips (either directly on the coals or in a smoke box) and preheat to a medium-low, about 265 degrees F. Add the pork and cook until the internal temperature reaches 140 degrees F, about 1 hour. Let rest for 10 to 15 minutes and then slice into thin slices.
For the beer cheese squash soup: Heat a heavy-bottomed soup pot over medium-high heat. Add the oil, onion, garlic and 1/8 teaspoon salt and cook, stirring, until soft but no color, 4 to 5 minutes. Add the squash and chicken stock, bring to boil, and then reduce to a simmer. Cook until very soft, 8 to 10 minutes. Add the red pepper flakes. Let cool slightly, then carefully puree the soup in batches in a blender. Alternatively, use an immersion blender to puree. Return the soup to the pot, add the beer, half-and-half, nutmeg and remaining salt. Stir in the Cheddar and jalapeno Jack until melted.
For the coleslaw: Mix the cabbage and carrot in a medium bowl. In a small bowl, combine the mayo, chives, red wine vinegar, celery seeds, salt and black pepper. Toss with the grated vegetables. Set aside.
For the sandwich: Preheat a cast-iron griddle over medium heat.
To assemble: Lightly brush the inside of a roll on the top and bottom with some softened butter. Cook cut-side down on the griddle over medium heat until toasted, about 3 minutes. Spread 2 teaspoons mustard on the bottom piece. Add 2 ounces sliced smoked pork, 1 slice bacon, 1 slice ham, 2 dill pickle slices and 1/2 cup coleslaw. Add the top bun.
Brush the top and bottom of the bun lightly with some softened butter, add the sandwich to the griddle, pressing to flatten with a spatula or another cast-iron pan, and cook until crisp and beginning to brown, about 4 minutes per side. Remove from the heat to a cutting board. Repeat to form 4 sandwiches total.
Ladle 1/4 cup beer cheese squash soup into 4 small ramekins. Garnish each with 2 teaspoons Cheddar and serve hot on the side with the sandwiches for dipping.
Reserve the remaining soup and pork tenderloin for another use.
When blending hot liquid, first let it cool for five minutes or so, then transfer it to a blender, filling only halfway. Put the lid on, leaving one corner open. Cover the lid with a kitchen towel to catch splatters, and pulse until smooth.