Preheat oven to 325 degrees. Spray a 13 x 9-inch baking pan with cooking spray.
Place cubed brioche bread on a baking sheet and bake 10 minutes to lightly toast.
Place half the brioche bread cubes in the prepared baking pan.
Scatter approximately 1/3 of the asparagus stems over the bread (reserve tips for the top).
Scatter half the coarsely chopped artichoke hearts over the bread and asparagus.
Scatter half the diced ham over the bread, asparagus and artichokes.
Sprinkle with half the cheese.
Repeat layers with remaining bread cubes, asparagus stems, artichoke hearts and ham followed by the rest of the cheese.
Scatter asparagus tips over the top.
Whisk eggs and milk or half-and-half together in a bowl until well blended along with salt and black pepper.
Pour custard over the layers. (At this point, strata may be covered with plastic wrap and refrigerated overnight.)
Bake 1 hour or until puffed and golden around the outside.
Let rest 5-10 minutes. Cut into squares and serve.