In a small bowl, whisk together the balsamic vinegar, extra virgin olive oil, dijon mustard, rice vinegar, grated lemon peel, minced garlic, soy sauce or tamari, and a pinch of kosher salt and black pepper until fully emulsified. Place the boneless, skinless chicken thighs in a glass container or resealable silicone bag. Pour the marinade over the chicken, ensuring each piece is fully coated. Cover and refrigerate for at least 30 minutes, or up to 3 hours for deeper flavor.
Preheat the grill to medium-high heat, about 400°F, and make sure the grill grates are clean. Remove the marinated chicken thighs from the refrigerator and place them directly over the flame. Grill for 3-4 minutes per side (adjusting for thickness), flipping once, until the internal temperature reaches 165°F. Let the grilled chicken rest for 5-10 minutes before slicing or serving. I like to let the chicken rest fully, as it helps keep the meat juicy and tender.
While the chicken rests, place the whole cherry tomatoes in a medium bowl. Drizzle with olive oil and season with a bit of kosher salt, tossing to coat evenly. Transfer the tomatoes onto a small grill pan and grill over direct heat, stirring occasionally, until the skins are slightly charred, about 5 minutes. Add the mozzarella pearls to the pan, then close the grill for 1-2 minutes just to gently warm the cheese—watch closely so the mozzarella doesn't melt.
Divide the prepared rice evenly among 4 serving bowls. Top with the rested grilled chicken thighs (from Step 2), charred tomatoes, and warmed mozzarella (from Step 3). Garnish each bowl with fresh basil leaves and finish with a generous drizzle of balsamic glaze. For a truly fresh and fragrant finish, I like to tear the basil leaves just before serving so they release more aroma.