In a large nonstick skillet, over medium heat, melt butter. Add asparagus and onion, cook until tender, about 5 minutes. Remove from heat.
In a large bowl, combine bread, ham, tomatoes and asparagus mixture; toss to combine.
In a separate large bowl, whisk eggs and milk, until combined. Add Parmesan cheese, nutmeg, salt and pepper. Pour over bread mixture and toss gently to coat. Cover and refrigerate for 1 hour or overnight.
Preheat oven to 350°F (180°C).
Generously coat a 12-cup muffin tin with nonstick cooking spray or oil. Divide the strata mixture between the muffin cups. Top with cheddar cheese.
Bake until puffed and golden, about 30 minutes. Remove from oven and let rest for 5 minutes before removing. Serve warm.
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