1. In a medium bowl, whisk together the mayonnaise, giardiniera brine, beef bouillon paste, garlic, celery salt, rosemary (if using), ½ teaspoon kosher salt, and ¼ teaspoon black pepper until fully blended. Cover and refrigerate for 30 minutes up to 1 day. Just before serving, mix a few tablespoons of the rested flank steak juices
2. Set up your grill for direct grilling over medium-high heat (550–660°F), creating a two-zone setup with an indirect cooking zone. A charcoal fire is recommended for best flavor.
3. Lightly coat the bell peppers with oil and season with salt. Place them over direct heat, close the lid, and grill for about 8 minutes, turning every few minutes until charred and softened on all sides. Transfer the peppers to indirect heat and grill for another 10 minutes until tender. Promptly transfer the charred peppers to a bowl, seal tightly with aluminum foil, plastic wrap, or a lid and let them steam for 10 to 15 minutes.
4. Gently peel away the blackened skin with your fingers. Slice the peppers open, then remove and discard the core and seeds. Avoid rinsing them with water to preserve their rich, smoky flavor. Slice into ¼-inch thick slices. Transfer to a bowl.
5. While the grilled flank steak rests for 10 minutes after cooking according to the recipe, butter the cut side of each roll. To ensure even browning, place the rolls buttered-side down over direct heat and press firmly with a grill press until lightly golden.Transfer them cut-side up to the indirect cooking zone, crown each bottom roll with 2 slices of cheese, and close the grill lid for about 30 seconds to let the cheese melt. Slather some of the steak sauce aioli on the cut side of the top half of the roll.
6. Assemble each sandwich by layering the sliced steak, generous heap of grilled peppers and giardiniera capping with the top rolls.
