Warm oil in a medium saucepan over medium heat.
Sauté onion, poblano, and jalapeño until softened, about 5 to 6 minutes.
Stir in garlic and cook until fragrant, about 30 seconds.
Sprinkle in cumin, chili powder, smoked paprika, and cornstarch. Cook 1 minute, stirring to coat.
Whisk in evaporated milk gradually and bring to a gentle simmer to thicken.
Lower heat. Add American and Monterey Jack in handfuls, stirring until smooth and melted.
Fold in green chiles, cilantro, and lime juice. Season with salt and pepper.
Adjust consistency with a splash of milk if needed; keep heat low to prevent breaking.
Spoon guacamole into a shallow bowl. Pour queso around it and top with pico, cotija, and hot sauce.
Serve hot with warm tortilla chips. Hold on low heat or in a small slow cooker.