Cut the white American cheese into 1/2-inch cubes so it melts evenly. Shred the Monterey Jack if not pre-shredded. Mince the jalapeño and garlic, draining the canned green chiles well.

Melt butter in a medium saucepan over medium heat. Add the jalapeño and garlic and cook for 1 minute until fragrant but not browned. Stir in the green chiles, cumin, and cayenne if using. Cook 30 seconds more.

Pour in the milk and bring to a gentle simmer, not a full boil. Reduce heat to low and add the cubed American cheese a handful at a time, whisking constantly until each addition melts completely before adding more.

Once all American cheese is melted and smooth, stir in the shredded Monterey Jack. Whisk until fully incorporated and the dip is silky with no stringiness, about 2 minutes. If too thick, thin with a splash more milk.

Taste and adjust salt as needed. The cheese is often salty enough on its own. Transfer to a serving bowl immediately and top with chopped cilantro. Serve warm with tortilla chips.