Whisk soy sauce, honey, sesame oil, Sriracha, garlic, and ginger puree together in a large bowl. Add chicken, cover, and refrigerate at least 4 hours to overnight.

Preheat an outdoor grill for medium heat and lightly oil the grate.

Thread marinated chicken thighs onto metal skewers and place on a platter, reserving marinade. Add squash, red onion, bell pepper, and mushrooms to the bowl with the marinade and stir until evenly coated.

Thread yellow squash, red onion, bell pepper, and mushrooms onto skewers and place on the platter with the skewered chicken.

Grill chicken and vegetable skewers for 20 minutes, turning after 10 minutes. Transfer skewers to a clean serving platter. Garnish with chopped cilantro.