For the spicy calamari and cucumber salad: Add the soy sauce, vinegar, honey, chili paste, garlic and ginger to a bowl and whisk to combine. Season with salt if needed. Divide the marinade between 2 medium bowls. Add the calamari to one of the bowls and the cucumbers to the other. Stir so that the calamari and cucumbers are fully coated in the marinade. Let marinate at least 30 minutes.
Preheat a grill pan over medium-high heat. Remove the calamari from the marinade and pat dry. Grill the calamari in batches until charred and tender, 1 to 2 minutes per side. Transfer to a cutting board and slice the tubes into 1/2-inch rings. Add the rings and whole tentacles to the bowl with the cucumbers and toss to combine.
For the spiced yogurt sauce: Whisk together the yogurt, chopped cilantro, coriander, garlic, caraway seeds, sumac, paprika and 1 teaspoon salt. The sauce can be made up to 1 day in advance and kept refrigerated.
To serve: Spread some of the yogurt sauce onto a shallow dish. Use a spoon to create a well in the center of the sauce so that the sides of the sauce are about 1/2-inch high. Add the marinated cucumbers and grilled calamari into the well. Garnish with cilantro leaves and black sesame seeds. Serve immediately.
English cucumbers can be substituted for Persian cucumbers in the same amount.