Not only should Musee du vieux Montreux be visited, but also Mila Cafe. You are offered to eat nicely cooked sandwiches and good salads. Google users awarded the rating of 4.4 to this restaurant.
The cafe only had one person working, and it took almost 20 minutes to get our breakfast order, which was a combination of premade salad and reheated frittata. The croissants and pain au chocolat were decent... So I would advise sticking to pastries. The cafe latte was also disappointing because the bottom was a layer of refrigerator temperature cold milk.