Wash and dry all produce. Preheat broiler or grill to high. Halve zucchini and squash lengthwise. Cut half the zucchini and half the squash lengthwise into long, 1/2-inch-thick planks. Cut the remaining halves into 4-inch-long and 1/2-inch-thick sticks, like fries. Mince 1 clove garlic (use other as you like). Pick mint leaves from stems; discard stems.

Zest lemon until you have 1 tsp zest, then cut into halves. In a small bowl, combine mayonnaise, zest, a pinch of garlic, and a squeeze of lemon. Season with salt, pepper, and more lemon juice and garlic (to taste).

Toss all zucchini and squash in a large bowl with a large drizzle of olive oil and sumac. Season with salt and pepper. If broiling, spread on a baking sheet in a single layer. Broil until golden brown, 10-15 minutes, flipping halfway through. If grilling, grill over direct heat until lightly charred and softened, about 10 minutes, flipping halfway through.

Cut cheese into thirds widthwise (parallel to the cutting board) to create 3 flat, wide slices. Set aside one slice for another use. If cooking indoors, heat a drizzle of oil in a medium pan over medium-high heat. Add remaining 2 cheese slices and cook until browned and melty, 1-2 minutes per side. If grilling, grill 2 cheese slices over direct heat until melty and grill marks appear, about 3 minutes per side.

Meanwhile, split buns in half and toast under broiler or on grill until golden brown, 2-3 minutes.

*Spread buns with aioli, then fill with cheese, a few mint leaves, and zucchini and squash planks. (TIP: Cut veggies as needed to fit in buns.) Roughly chop remaining mint and scatter over zucchini and squash sticks. Serve burgers with veggie sticks to the side.