In a medium bowl, whisk together the olive oil, lime juice, minced garlic, paprika, cumin (or chili powder), salt, and black pepper to make the marinade.
Add the chicken breasts to a large zip-top bag or shallow dish and pour the marinade over them. Seal or cover and refrigerate for at least 30 minutes (up to 4 hours) to let the flavors penetrate.
Preheat your grill or grill pan to medium-high heat and lightly oil the grates to prevent sticking.
Remove the chicken from the marinade and shake off any excess. Place the chicken on the hot grill and cook for about 6–7 minutes per side, or until the internal temperature reaches 165°F on a meat thermometer.
Once the chicken is cooked through, top each breast with a generous sprinkle of shredded mozzarella or Monterey Jack cheese. Close the grill lid for 1–2 minutes to let the cheese melt slightly.
Remove the chicken from the grill and immediately top with sliced avocado and halved cherry tomatoes.
Garnish with chopped cilantro, diced red onion, or a drizzle of balsamic glaze if desired. Serve hot with your favorite sides.