Step 1

Preheat the oven to 350°F. Line a sheet pan with foil and coat the foil with cooking spray. Arrange the bell pepper strips on the pan and coat with cooking spray; set aside.

Step 2

Coat a small Dutch oven, brasier, or cast-iron skillet (with tight-fitting lid) with cooking spray and heat over medium-high. Sprinkle the beef evenly with salt and pepper. Add the beef to the pan; cook until well browned, about 4 minutes on each side. Remove the beef from the pan. Add the carrots, onion, and celery to the pan; sauté for 3 minutes. Add the beef stock and thyme sprigs, scraping the bottom of the pan to release the browned bits; bring the liquid to a boil and return the beef to the pan.

Step 3

Cover the pan and place it in the oven with the bell peppers. Bake until the peppers are very tender and beef is well done, about 40 minutes. Remove from the oven. Remove the beef from the pan and place on a cutting board; allow the beef to cool for 15 minutes. Wrap the foil around the peppers to keep them warm.

Step 4

Turn the oven off and place the rolls inside for 3 to 5 minutes to crisp the exterior. While the beef cools, strain the cooking liquid through a strainer into a bowl. Return the strained liquid to the pan. Bring to a boil; reduce the heat to medium-low, and cook until slightly reduced, about 8 minutes.

Step 5

Using a very sharp knife, cut the beef as thinly as possible (cutting on an angle diagonal to the cutting board, instead of straight up and down, helps you get thinner slices). Add the sliced beef to the reduced cooking liquid. Split the rolls in half lengthwise, cutting to but not through the other side. Divide the bell peppers evenly among the rolls. Lift the beef out of the liquid with tongs and divide evenly among the rolls; top each sandwich with 2 tbsp giardiniera.

Step 6

Serving size: 1 sandwich and about ⅓ cup cooking liquid