If you wish, you can cut small slits in the surface of the meat every inch or so and stick slivers of fresh garlic into the meat. If you do this, leave the garlic out of the rub. Otherwise, mix the rub in a bowl. Sprinkle it generously on the meat and massage it in. There will be some left over. Do not discard it, we will use it in the juice. Let the meat sit at room temp for about 30 minutes. Meanwhile, preheat the grill or oven to 400°F. If you are cooking indoors, put a rack just below the center of the oven.
Pour the water into a 9 x 13" baking pan and heat it to a boil on the stove top. Dissolve the bouillon in the water. It may look thin, but it will cook down and concentrate during the roasting. Pour the remaining rub into the pan. Place a rack on top of the pan. Place the roast on top of the rack above the juice. Roast at 400°F until interior temperature is about 130°F for medium rare, about 30 minutes per pound (exact time will depend on the cut of meat, its thickness, and how well calibrated your oven is). This may seem long, but you are cooking over water and that slows things down. The temp will rise about 5°F more as it rests. Don't worry if there are people who won't eat medium-rare meat. The meat will cook further in step 5, and you can just leave theirs in the juice until it turns to leather if that's what they want. If you use a rotisserie on your grill, you can cut the cooking time in half because the spear and the forks holding it in place will conduct heat into the interior.
Beware. This recipe is designed for a 9 x 13" baking pan. If you use a larger pan, the water may evaporate and the juice will burn. If you have to use a larger pan, add more water. Regardless of pan size, keep an eye on the pan to make sure it doesn't dry out during cooking. Add more water if necessary
While the meat is roasting (mmmmm, smells sooooo good), cut the bell peppers in half and remove the stems and seeds. Rinse, and cut into 1/4" strips. Cook the peppers in a frying pan over a medium high heat with enough olive oil to coat the bottom, about 1 tablespoon. When they are getting limp and the skins begin to brown, about 15 minutes, they are done. Set aside at room temp.
Remove the roast and the juice pan. Take the meat off the rack and remove the rack. Pour off the juice, put the meat back in the pan, and place it in the coldest part of the refrigerator. Let it cool for a few hours, long enough for the meat to firm up. This will make slicing easier. Chill the juice, too, in a separate container. Slice the meat against the grain as thin as humanly possible, preferably with a meat slicer. My wife remembers that her family would cook the roast and take it to the butcher to slice on his machine. That's a good strategy if you don't have a meat slicer but it may be against your local health codes. If you don't have a slicer, use a thin blade and draw it along the meat. If you try to cut downor saw through the crust you will be cutting it too thick.
Taste the juice. If you want you can thin it with more water, or make it richer by cooking it down on top of the stove. In Chicago beef stands it is rich, but not too concentrated. Then turn the heat to a gentle simmer. Soak the meat in the juice for about 1 minute at a low simmer. That's all. That warms the meat and makes it very wet. You can't leave the meat in the juice for more than 10 minutes or else it starts to curl up, squeezes out its natural moisture, and toughens. If you go to a beef stand and the meat is really curly, they have committed a mortal sin. At Mr. Beef, for example, I watched them take a handful of cooked beef and dump it into the juice every time they took out enough for a sandwich. This also enriches the juice with meat protein and seasoning from the crust.
To assemble the sandwich, start by spooning some juice directly onto the bun. Get it wet. Then lay on the beef generously. Spoon on more juice (don't burn your hand). Top it with bell pepper and, if you wish, giardiniera. If you want it "wet", dip the whole shootin' match in juice. Be sure to have plenty of napkins on hand.
NOTE: Italian giardiniera is also called "sottaceti", meaning "under vinegar", a common term for pickled foods. It is typically eaten as an antipasto or with salads.
In the United States, giardiniera is commonly available in traditional or spicy varieties, and the latter is sometimes referred to as "Hot Mix".
In the cuisine of Chicago, giardiniera is used as a condiment, typically as a topping on Italian beef sandwiches.