To make the chicken, combine the yogurt, garlic, thyme leaves, Zesty Seasoning, lemon zest and juice, olive oil, salt, and pepper in a large bowl. Whisk well.
Add the cubed chicken to the marinade and stir well to coat. Cover the bowl with plastic wrap and marinate in the refrigerator for at least 30 minutes and up to 2 hours.
While the chicken marinates, make the yogurt dip. In a medium bowl, mix together the yogurt, Zesty Seasoning, lemon zest and juice, olive oil, basil, salt, and pepper. Set aside.
Once chicken has marinated, thread 2 to 3 pieces of chicken lengthwise onto a wooden skewer. Leave at least 1 inch of space at the bottom of each skewer so you can maneuver them while cooking. Repeat with the remaining chicken.
Heat a grill pan over medium-high heat. Brush a few teaspoons of vegetable oil on the pan. Season the chicken skewers with salt and pepper and add a few at a time to the grill pan, ensuring the skewers are laid perpendicular to the grill lines. Grill until the chicken is cooked through, 2 to 3 minutes per side. Set the cooked skewers on a plate or tray while you cook the rest.
Arrange the skewers on platter and serve with the yogurt dipping sauce.