In a large skillet with a lid, heat olive oil over medium heat.
Add the chopped onion and cook for 3 to 4 minutes, stirring occasionally, until softened.
In a medium bowl, toss the chicken with taco seasoning until evenly coated.
Add the chicken to the skillet and cook for 4 to 5 minutes, stirring occasionally, until lightly browned on the outside.
Stir in the garlic and cook for 1 minute.
Add the diced tomatoes with green chilies (with juices), tomato sauce, chicken broth, black beans, and rinsed rice. Stir to combine.
Bring to a gentle boil, then reduce heat to medium-low. Cover and simmer for 18 to 22 minutes, stirring once halfway through, until the rice is tender and most of the liquid is absorbed.
Remove the lid and sprinkle both cheeses evenly over the top. Cover for 2 to 3 minutes to melt, or broil for 1 to 2 minutes until bubbly and lightly golden.
Remove from heat. Garnish with cherry tomatoes and cilantro.
Serve warm with sour cream and a squeeze of lime.