Toss mushrooms and flour together in a bowl until coated.

Heat olive oil in a large skillet over medium heat; cook and stir coated mushrooms until browned, 5 to 10 minutes.

Combine potatoes, carrots, onion, garlic, and rosemary in a large stockpot; top with cooked mushrooms. Stir beer, vegetable broth, tomato paste, Worcestershire sauce, salt, and pepper into vegetable mixture. Cover stockpot and cook over medium-high heat until vegetables are tender and soup flavors have blended, about 1 hour.

Whisk water and cornstarch together in a bowl until dissolved; stir into soup until thickened, about 3 minutes.

Set oven rack about 5 inches from the heat source and preheat the oven's broiler.

Arrange bread slices on a baking sheet and sprinkle Swiss cheese over each slice.

Broil bread in the preheated oven until cheese is melted and golden, about 1 minute.

Ladle soup into bowls using a slotted ladle; top each with a bread slice. Ladle broth over bread until saturated.