Place the bread slices on a baking sheet and broil them until lightly toasted—I've found that broilers vary in intensity, so check every 2 minutes or so to make sure the bread doesn't burn. Flip the bread over and broil the other side. Remove from the oven and set aside.
Heat the olive oil in a large skillet over medium heat. Add the shallots and cook, stirring often, for about 4 minutes, until softened and just beginning to turn golden brown. Increase the heat to medium-high and add the mushrooms; cook for about 5 minutes, or until the mushrooms release their juices and it cooks off. Stir in the asparagus, thyme and vinegar; cook for about 3 minutes, or until the asparagus rounds are tender-crisp. Season to taste with salt and pepper.
Divide the vegetable mixture onto the bread. (You may have some leftover depending on the size of your bread.) Top the vegetables with the shredded cheese, then return the baking sheet to the oven and broil until the cheese has melted and is beginning to bubble and brown in places (checking every 2 minutes again to make sure they don't burn). Serve immediately.
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