Step 1

Position racks in top third and center of oven; preheat to 425°. Toss carrots, bell peppers, onion, red pepper flakes, and 3 Tbsp. oil on a rimmed baking sheet. Roast on center rack, tossing occasionally, until vegetables are tender and well browned (it’s okay if some spots are beginning to look burned; this will give the romesco a deeper flavor), 25–30 minutes.

Step 2

Meanwhile, separate broccolini florets from stalks; thinly slice stalks. Transfer florets and stalks to a clean rimmed baking sheet. Add garlic and ¼ cup oil, season with salt, and toss to combine. Roast on top rack, tossing halfway through, until florets are charred and stems are cooked through but still have some crunch, 16–18 minutes. Pour 2 tsp. vinegar over and toss to coat; season with salt.

Step 3

Transfer carrot mixture to a food processor. Add 2 Tbsp. vinegar and pulse until smooth. With motor running, gradually drizzle in remaining ½ cup oil and process, scraping down sides as needed, until emulsified; season with salt.

Step 4

Slice rolls in half lengthwise. Arrange cut side up on a rimmed baking sheet and toast until light golden brown, about 3 minutes.

Step 5

Divide romesco between tops and bottoms of rolls. Pile broccolini on bottom halves. Top broccolini with cheese. Toast sandwiches, open-faced, until cheese is just melted, about 2 minutes. Close sandwiches and cut crosswise before serving.