Start the rice: rinse the uncooked rice until the water runs clear. Combine the rice, 1 3/4 cups water and 1/2 tsp salt in a small saucepan. Bring to a boil, reduce to low, cover and simmer 15 minutes. Remove from heat and let sit, covered, 5 minutes while you prepare the rest.
Season and sear the chicken: pat the chicken dry and rub with olive oil, cumin, lime zest, minced garlic, 1/2 tsp salt and black pepper. Heat a large skillet over medium-high heat. Sear the breasts 2 minutes per side until lightly golden to build flavor.
Poach in salsa verde: pour 1 cup salsa verde into the skillet with the seared chicken, bring to a gentle simmer, then cover and cook 10–12 minutes, or until internal temperature reaches 165°F. Reduce heat if the salsa is bubbling too vigorously.
Shred and finish the chicken: transfer chicken to a cutting board and let rest 3 minutes. Shred with two forks, then return shredded chicken to the skillet and stir into the salsa for 2–3 minutes to let the sauce reduce and cling to the meat.
Warm the beans and corn: while the chicken finishes, place the drained beans and thawed corn in a small saucepan with a squeeze of lime juice and a pinch of salt. Heat over medium until warmed through, about 3–4 minutes.
Make the yogurt drizzle: whisk the Greek yogurt with the extra 2 tablespoons salsa verde until smooth and spoonable. Taste and add a squeeze of lime if you like it tangier.
Assemble the bowls: divide the cooked rice among four bowls. Top each with a portion of shredded salsa verde chicken, warmed black beans and corn, sliced avocado, a sprinkle of cheese and chopped cilantro.
Finish and serve: dollop the yogurt drizzle over each bowl, squeeze remaining lime juice overtop if desired, and serve immediately.