Melt butter in a skillet over medium heat. Add onion and red bell pepper; cook for 2 minutes. Add garlic and cook for 1-2 additional minutes until fragrant. Add green chiles, cumin, chili powder, and salt. Stir to combine. Add rice and toast for 2 minutes.
Slowly pour in vegetable stock and milk, stirring to combine. Bring back up to a simmer, cover, and lower heat so rice continues to slowly simmer. Slowly simmer for 15-20 minutes until liquid is mostly absorbed and rice is tender.
Remove from heat and stir in cheese in batches, mostly incorporating the cheese before adding more (it's easier to mix together in batches instead of adding it all at once). Top with cilantro and a squirt of a lime wedge. Serve immediately.
Notes
I use whatever white rice I have on hand - (basmati or jasmine even, although they're not really as authentic to Mexican rice). Adjust cook time by a few minutes as needed depending on which rice you use.