In a small bowl, combine 8 tablespoons softened butter with 1 teaspoon ground cumin and 2 tablespoons chopped cilantro; set this flavored butter aside.
Heat a large deep skillet over medium and brown 8 oz breakfast sausage, breaking it into small pieces as it cooks.
If the skillet looks dry, add 1 tablespoon salted butter, then add 1/2 finely diced yellow onion and sauté with the sausage until the onion is tender, about 5 minutes; reduce heat to medium-low.
Gradually melt 1 lb cubed Velveeta (or American cheese) into the pan, stirring until smooth, then add 4 oz shredded Monterey Jack cheese a little at a time until incorporated.
Stir in 10 oz drained diced tomatoes with green chiles and 4 oz diced green chiles; the queso will be thick—stir in 6 oz evaporated milk to thin slightly, then season with salt and pepper to taste; keep covered and warm over low heat.
Bring a large pot with about 12 cups of water to a boil, add approximately 2 tablespoons white vinegar, then reduce to a gentle simmer for poaching.
Crack eggs individually into small bowls and slide them into the simmering water two at a time; cook each batch for 3 minutes, then remove with a slotted spoon and drain on paper towels; repeat until all 8 eggs are poached.
Toast the 4 English muffins (halved) and spread each half with about 1 tablespoon of the prepared cumin-cilantro butter.
Place one poached egg on each muffin half, spoon warm chile con queso over the eggs, then finish with avocado, pico de gallo, sour cream, cilantro, pepper, and chives as desired; serve immediately.
