Meaty portobello mushrooms have just the right amount of earthy, savory flavor and the perfect texture to replace beef in these vegetarian French dip sandwiches.
Heat 1 tablespoon oil in a large cast-iron or heavy skillet over high heat. Add onion and cook, stirring frequently, until lightly browned, 3 to 4 minutes. Add garlic and cook until fragrant, about 1 minute. Transfer to a medium bowl.
Add 1 tablespoon oil to the pan. Add portobellos and cook, stirring occasionally, until browned, 5 to 6 minutes. Transfer to the bowl. Add the remaining 1 tablespoon oil and oyster (or shiitake) mushrooms to the pan. Cook, stirring occasionally, until browned, 5 to 6 minutes. Return the portobellos and onion to the pan. Stir in mustard and thyme; cook for 1 minute. Stir in broth and sherry and simmer for 5 minutes.
Position a rack in upper third of oven; preheat broiler to high.
Scoop out most of the inside of the rolls (reserve for making breadcrumbs, if desired). Place the rolls cut-side up on a baking sheet. Using a slotted spoon, divide the mushroom mixture among the bottom halves of the rolls (reserve the sauce). Top with cheese.
Broil until the cheese is bubbly and lightly browned, 1 to 2 minutes. Place the top halves of the rolls on the sandwiches and serve with the sauce.