Melt butter in a pot or Dutch oven over medium heat.
Add mushrooms and shallots. Cook, stirring occasionally, until tender, about 6-8 minutes. Stir in garlic until fragrant, about 1 minute.
Whisk in flour until lightly browned, about 1 minute. Stir in wine (if using), scraping any browned bits from the bottom of the pot.
Stir in chicken stock, thyme, and bay leaves; season with salt and pepper, to taste. Bring to a boil; reduce heat and simmer for 10 minutes.
Stir in potatoes; cover and simmer until tender, about 10 additional minutes. Remove from heat; stir in cream and parsley.
In the meantime make the sandwich: Mix the mayo and garlic in a bowl. Season with salt and pepper to taste. Toast the bread. Then spread the mayo mixture on one side of each slice.
Layer with arugula, pastrami, and cheese.
Cut in half, and serve with pickle, soup, crackers, and sparkling water.