Peel white asparagus with a vegetable peeler from top to bottom, starting below the tips. Cut off any woody ends with a knife.
Place asparagus spears and salt in a large, wide saucepan and pour in enough water to just cover. Bring to a boil. Reduce heat; simmer until spears are tender and easily pierced with a knife, 15 to 25 minutes. Drain.
Set oven rack about 6 inches from the heat source and preheat the oven's broiler.
Toast bread slices; arrange on a baking sheet. Spread butter over warm bread.
Lay 1/2 slice ham on top of each slice of bread; trim excess on all sides. Cut asparagus spears in half; divide evenly over ham. Cover asparagus spears with Edam cheese.
Place under the preheated broiler until cheese starts to bubble and brown slightly, about 3 minutes.