Make the White Bean Purée: Reserve ¼ cup of the beans and set aside. Place remaining beans in the work bowl of a food processor along with water, olive oil, lemon juice and garlic. Process until smooth.
Add rosemary and reserved beans and pulse until beans are slightly broken down and rosemary is combined into the mixture. Season to taste with salt and pepper.
Make Balsamic-Glazed Mushrooms: Heat olive oil in a 10-inch nonstick skillet over medium heat. Add mushrooms and a pinch of salt. Increase heat to medium-high and sauté mushrooms until tender and golden brown, approximately 7-9 minutes.
Add vinegar and crushed red pepper flakes. Reduce heat to medium low. Continue to cook until vinegar forms a light glaze over the mushrooms, about 1 minute. Season to taste with salt. Set aside to cool slightly.
Assemble sandwiches: Toast bread and slice on the diagonal into triangle shapes. Rub garlic clove over the surface of one side of each piece of toast.
Spread a generous layer of White Bean Purée over each slice of toast (there will be some puree left over).
Scatter arugula on top of the purée. Place Balsamic-Glazed Mushrooms on top of arugula, dividing the mushrooms evenly between the 4 sandwiches. Serve.