Place the broth, horseradish, garlic, rosemary, and thyme in a large pot. Stir to combine and bring to a simmer over medium heat.
Once simmering, add the pastrami to the pot.
Reduce the heat to low, cover, and cook for 15 minutes.
Meanwhile, melt the butter in a large skillet over medium heat.
Once melted, add the onions, and saute them until they begin to soften and brown, about 5 minutes.
Deglaze the skillet with beer, stir to combine with the onions, and bring to a simmer.
Simmer until the beer has almost completely evaporated, about 10 minutes.
Add the Dijon mustard, grain mustard, and honey to the skillet and stir to combine.
Continue to cook for 2 minutes until the onions become jammy. Remove from the heat and set aside.
Place bread slices on a sheet of wax paper and spread butter over one side.
Flip 4 of the bread slices over, butter side down.
Divide the Cooper Sharp cheese between the 4 slices.
Top the cheese with hot pastrami (let any excess au jus drip off the pastrami before transferring to the bread). Reserve the au jus.
Top the pastrami with the onion mixture, spreading it out evenly.
Top the onions with the cheddar and fontina cheese slices.
Cap the sandwiches with the remaining bread slices, butter side-up.
Preheat a panini press, panini pan, or grill pan over medium heat.
Once hot, add the sandwiches. Firmly and evenly press the sandwiches (using either a sandwich press or a heavyweight pot or pan).
Toast for 2 minutes or until golden brown on one side. Flip, press, and repeat on the other side until the sandwiches are toasty and the cheese has melted.
Ladle the au jus between serving cups and serve alongside the sandwiches for dipping.