Make the yogurt sauce: In a bowl, combine all ingredients and mix to incorporate.
Heat a grill pan over medium heat. Slice each squash in half lengthwise. If the squash are quite fat, slice them into thirds, and if they are more than 6 inches long, half crosswise too. Brush generously with olive oil and sprinkle with salt.
Place sliced squash, cut-side down, on grill pan and cook undisturbed until dark grill marks appear, 4-5 minutes. Flip and cook until skin side is also charred, 5 minutes more. Remove finished pieces and set aside. Alternatively, if you are already firing up your grill, you can grill the squash until it has char marks on both sides.
To assemble, spoon yogurt sauce over a serving platter. Arrange grilled squash on top. Sprinkle with sumac, dried mint, pepper, toasted nuts and herbs.