In a large bowl, combine beans, pepper Jack, chili powder, and cumin.
Place tortillas on a heatproof plate and cover with a damp paper towel. Microwave until warm, about 15 seconds.
Place about 2 tablespoons bean mixture in center of tortilla. Top with cheddar and tightly roll up.
Into a large, high-sided skillet, pour oil to a depth of about ½". Heat over medium heat until oil is shimmering (an instant-read thermometer should register about 350°).
Working in batches, fry rolled tortillas seam side down until golden, 2 to 3 minutes (flour tortillas may take 4 to 5 minutes). Flip and fry second side until golden, 1 to 2 minutes more. Using tongs, transfer to a paper towel-lined plate. Repeat with remaining tortillas, adjusting heat as needed.
In a medium bowl, combine green chiles, sour cream, cilantro, garlic, and lime juice; season with salt and pepper.
Serve taquitos warm with dipping sauce alongside.
Make ahead: Sour cream sauce can be made 1 day ahead. Store in an airtight container and refrigerate.
