Preheat the broiler and line a baking sheet with aluminum foil. Arrange the tomatillos, Roma tomatoes, whole garlic cloves, and serrano peppers on the tray. Drizzle lightly with olive oil. Broil for 8–10 minutes, or until the skins of the tomatoes and tomatillos split and char, and the peppers are blistered. Rotate and continue roasting an additional 8 minutes to ensure even charring. Remove from the oven and allow to cool slightly.

Once cooled, peel the tomatoes and discard the skins. Remove the stems and seeds from the serrano peppers. In a blender or food processor, combine the roasted tomatillos, peeled tomatoes (reserve two for later use), serrano peppers, roasted garlic, ½ teaspoon salt, 1 tablespoon chopped cilantro, and 1 tablespoon lime juice. Blend until smooth. Set aside.

Finely dice the remaining two Roma tomatoes and reserve for later use, along with the diced Hatch green chiles.

In a large, heavy-bottomed saucepan, melt the butter over medium heat. Add the minced garlic and sauté for approximately 30 seconds, or until fragrant. Stir in 1 cup of half-and-half, the diced tomatoes, and the diced Hatch green chiles. Bring to a gentle simmer.

Gradually add the cubed Velveeta Queso Blanco and shredded sharp cheddar cheese. Stir continuously until the cheeses melt completely and the mixture becomes smooth and creamy. Incorporate the lime juice, ½ teaspoon kosher salt, and half of the prepared salsa verde. Adjust by adding more salsa if additional heat is desired.

Remove the queso from the heat and allow it to rest for at least 1–2 hours. This resting period enhances the flavors by allowing the roasted vegetables to infuse into the cheese mixture. The queso may be refrigerated and gently reheated as needed. Thin with additional half-and-half if necessary during reheating.

Place a generous scoop of guacamole into a serving bowl. Ladle the warm queso over the guacamole. Garnish with crumbled cotija cheese, chopped cilantro, and hot sauce if preferred. Serve immediately with tortilla chips.