Preheat your oven to 450°F (232°C).
In a 4–8 qt. pot, heat the oil over medium-high heat. Add the sliced carrot, sliced onion, and both halves of the garlic bulb. Sautee until things just start to brown.
Add 2 quarts of good quality beef stock to the pot and bring to a high simmer. Cook 30 minutes. Then strain out the veggies and keep the gravy in the pot.
While the gravy is heating and simmering, wrap the hoagies in foil and place them in the oven for 8–10 minutes to warm and soften.
Check the temperature of the gravy with your Thermapen ONE. If the temperature is above 140°F (60°C) whisk it vigorously to cool it down more quickly.
Once the gravy lands on 140°F (60°C), adjust the heat on your stove to try to keep it there. This will be kind of like deep frying, as we add things to the pot it'll cool, so we need to add more heat.
Take 5 ounces of the sliced meat and submerge it in the gravy with some tongs. Let it sit for a liittle while, about half a minute, while you slice one of the rolls most of the way through, but not all the way through.
With the tongs, grab the roast beef from the gravy and lay it in the bun, either keeping or shaking off the excess gravy.
Top with rosted peppers and/or giardiniera, then, if you want, use the tongs to dunk the whole sandwich in the gravy. How long you can take to do this will depend on the structure of your bread! Too long and it'll fall apart.
Devour your sandwich, dripping with juice, over a plate. Repeat.