In a medium bowl whisk honey, lime juice, olive oil, garlic, salt, pepper, and cumin if using. Add the chicken and turn to coat. Cover and refrigerate 20–30 minutes. Do not marinate longer than 1 hour or the lime can start to toughen the surface.
Rinse the rice under cold water until the water runs mostly clear. Combine rice, water or broth, and salt in a medium saucepan. Bring to a boil, then reduce heat to low, cover, and simmer 15–18 minutes until the liquid is absorbed and grains are tender. Remove from heat and let sit covered 5 minutes. Fluff with a fork, then stir in lime juice and cilantro.
Heat a large skillet or grill pan over medium heat. Remove chicken from marinade (discard leftover marinade) and cook 5–7 minutes per side until the internal temperature reaches 165°F and the exterior shows light golden edges. Rest the chicken 5 minutes, then slice against the grain into thin strips.
Just before serving, halve the avocados, remove pits, and slice or lightly mash. Toss gently with 1 tablespoon lime juice and a pinch of salt so the pieces stay bright green.
Use a round mold, measuring cup, or simply layer free-form on plates. Start with a scoop of rice, press lightly, add sliced chicken, then top with avocado. Finish with cilantro leaves and a lime wedge. Serve immediately while the chicken is warm and the avocado is cool.