Cooking vegetables over fire releases their natural sugars and deepens their flavors. We like them simply seasoned with salt and pepper, but dry rub works well, too.
Wash the vegetables and lay the broccolini and asparagus out on paper towels. Cut the carrots and leeks in half lengthwise, and quarter the beets, peppers, and onion. Prepare a large bowl of ice water and bring a large pot of salted water to a boil.
Working with one vegetable at a time, boil the carrots, leeks, and beets for 3 to 5 minutes. Transfer to the ice water for about 1 minute. Pat dry with paper towels. (Cook and chill the beets last, so they don’t turn the other vegetables pink.)
Combine all the vegetables in a large bowl or on a baking sheet. Drizzle with olive oil, season well with salt and pepper, and toss to coat.
Set up the cooker for direct cooking: Open the top and bottom vents. Pile 2 pounds of the charcoal in the bottom. Load a charcoal chimney one-quarter full of charcoal and light it. When the coals in the chimney are glowing, dump them on top of the pile already in the cooker and close the lid. Adjust the vents as necessary to establish a steady temperature between 350° to 375° for direct grilling (See pages 84 and 77 for how to best assess and monitor cooker temperature, and how to determine your target temperature). Open the cooker and spread the vegetables evenly over the grate and cook for 3 to 4 minutes, or until the edges begin to char. Flip and cook for 3 to 4 minutes on the other side, or until the peppers, broccolini, asparagus, and onion are charred at the edges and slightly soft (pull individual pieces off as they reach this point). Transfer to a large serving platter. The beets, carrots, and leeks will take a few minutes longer; don’t let them get too soft.
Garnish the platter with herb sprigs. Serve the vegetables hot or at room temperature. Leftovers can be refrigerated and eaten cold, or chopped for salads.