Butter 8-by-8-inch baking dish and set aside. Heat butter over medium heat. When it foams, add scallions, and cook until just softened, about 1 minute. Add asparagus and cook until bright green and beginning to soften, about 2 minutes; remove from heat.
Whisk eggs, half-and-half, salt, cumin, and pepper in a large bowl until evenly combined. Arrange half of the bread in bottom of prepared dish. Sprinkle with 1/2 cup of the cheese and half each of jalapenos, cilantro, and asparagus mixture. Pour over half of egg mixture. Repeat layering with remaining bread, another 1/2 cup of the cheese, and remaining chilies, cilantro and asparagus mixture. Pour remaining egg mixture over top.
Sprinkle strata with another 1/4 cup of the cheese and push to ensure bread is submerged in egg custard. Cover with plastic wrap, put a plate and a 1-pound weight on top (a large can of tomatoes or rice in a re-sealable plastic bag works well), place in refrigerator for 1 to 12 hours.
When ready to bake, heat oven to 325 degrees F and arrange rack in the middle. Let strata sit at room temperature while oven heats, at least 20 minutes.
Sprinkle strata with remaining 1/4 cup of the cheese and bake until puffed and edges pull away slightly, about 40 to 45 minutes. Cool for at least 5 minutes before serving. Serve with some cilantro, sour cream, and avocado, as desired.