Place oven racks in top third and center then preheat to 425°F.
Toss carrots, bell peppers, onion, red pepper flakes, and 3 tablespoons olive oil on a sheet pan. Roast on center rack, tossing occasionally, until vegetables are tender and well browned, about 25–30 minutes.
Once browned, transfer roasted carrots, bell peppers, onion, red pepper flakes into a food processor and season with salt. Add 2 tablespoons sherry vinegar and process. While processing, gradually drizzle in remaining ½ cup oil and process, scraping down sides as needed, until emulsified. Set aside.
Separate Broccolini florets from stalks; thinly slice stalks. Transfer florets and stalks to a separate sheet pan. Add chopped garlic to Broccolini and top with ¼ cup oil, then season with kosher salt. Toss to combine.
In oven, roast Broccolini on top rack, tossing halfway through, until florets are charred and stems are cooked through but still maintain their crunch, about 16–18 minutes.
Slice rolls in half lengthwise. Arrange cut side up on a rimmed baking sheet and toast until light golden brown, about 3 minutes.
Divide sauce between tops and bottoms of rolls. Pile Broccolini on bottom halves. Top Broccolini with cheese. In oven, toast sandwiches open-faced, until cheese is just melted, about 2 minutes. Close sandwiches and cut crosswise before serving.
Recipe provided by Rooster Soup Co. in Philadelphia, PA.