Toss the shredded cheddar and Monterey Jack together in a medium bowl; set aside.
In another medium bowl, whisk the eggs, whole milk, smoked paprika, cayenne, kosher salt, and black pepper until combined.
Drain the canned green chiles and place them on a cutting board; cut each chile in half lengthwise.
If using chorizo, brown the required amount in a medium sauté pan over medium heat until cooked through, then drain and set aside.
Grease an 8x8-inch baking pan (or two 4x8-inch pans for half-and-half) with butter or nonstick spray.
Arrange about half the chile halves in the bottom of the baking dish (roughly 6 halves for an 8x8), then lay four slices of bread over the chiles.
For a full-vegetarian strata: pour half the egg mixture over the bread and chiles, sprinkle half the cheese, then repeat the layers with the remaining chiles, bread, egg mixture, and cheese; dust with a little smoked paprika.
For a full-meaty strata: after placing the bread, sprinkle half the cooked chorizo over the bread, pour half the egg mixture, top with half the cheese, then repeat the layers (chiles, bread, chorizo, eggs, cheese); dust with smoked paprika.
For a half-and-half version using two pans: distribute chiles and bread between the two pans (about 3 chile halves and 2 slices bread per pan). Add cooked chorizo to the meaty pan only. Divide the egg mixture and cheese roughly evenly between the pans and finish with a sprinkle of smoked paprika.
Cover tightly with plastic wrap or foil and refrigerate overnight or for at least 3 hours to allow the bread to soak.
Uncover and bake at 325°F (160°C) for 40–50 minutes, until the strata is puffed, golden, and set in the center.
Let cool a few minutes, then serve with desired optional toppings.