Bring broth and water to a boil in a 4-quart pot. Add asparagus and cook uncovered until crisp-tender, 3-4 minutes. Transfer asparagus with a slotted spoon to a large bowl of ice water to stop cooking, then drain and pat dry. Maintain broth at a low simmer, covered.
Heat oil with 1 tbsp butter in a 4-quart heavy saucepan over med-high heat until foam subsides, then saute mushrooms for about 4 minutes until browned, stirring occasionally. Season with salt and pepper, then transfer to a bowl.