Marinate the chicken. Combine chicken with olive oil, lemon juice, garlic, tomato paste, paprika, salt, black pepper, cumin, coriander, and allspice until well mixed. Let sit for at least 1 hour or overnight in the fridge.
Meanwhile, make zucchini babaghanoush. Place zucchini on a sheet pan and broil on high for 45 minutes - 1 hour or until deeply browned on all sides. Rotate zucchini throughout to brown evenly. Remove from heat and let cool slightly.
Preheat oven to 425 degrees F.
When zucchini is cool enough to handle, peel off skin and discard. Place zucchini in a colander and let sit for 10 minutes to drain excess liquid.
Mash zucchini with garlic and lemon juice in a large bowl. Slowly add olive oil while mixing. Add Greek yogurt, stirring until combined. Season with salt and pepper and set aside.
Skewer chicken alternating with red onion. Place skewers on a sheet pan and roast in the oven for 10 minutes. Flip over the skewers and roast for another 5 minutes or until chicken is cooked and lightly charred.
Remove chicken from the oven and place on a plate. Sprinkle fresh dill on top and serve with zucchini babaghanoush, lemon wedges, and pita bread, if desired.