Combine the seasoning, olive oil, red wine vinegar, salt, pepper, paprika, thyme, oregano, garlic, and lemon juice in a bowl; whisk to combine. Add in the chicken. Cover and chill at least 1 hour.

Prepare grill over medium-high heat. Thread chicken pieces on skewers, dividing equally. Discard marinade in bowl.

Grill chicken until golden brown and cooked through, turning skewers occasionally, 10 to 12 minutes total. Transfer skewers to platter. Serve with basil vinaigrette and extra lemon wedges.