In a medium non-stick skillet over medium heat melt 2 tablespoon butter until it foams. Add in the mushrooms and shallots cooking for 3-5 minutes or until softened. Add in the Worcestershire, salt and stir. Add in the arugula and mix until it wilts. ~2-3 minutes. Remove from the heat and set aside.
While the mushroom mixture is cooling place the ricotta, 1/4 cup of Parmesan and torn prosciutto pieces in a bowl. Mix to combine. Add the mushroom mixture and mix just until incorporated.
In a pie dish whisk together the eggs, milk and remaining Parmesan cheese until well blended and slightly frothy. Take a slice of bread and using a rolling pin, roll to flatten down. You can choose to remove the crusts. Take 2 tablespoon of mixture and spread it out from end to end at the side closest to you. Grabbing the edge of the bread closest to you roll up the bread and gently press down as you roll to help seal. Continue on with the remaining slices or until all of the mixture is gone.
Take a large non-stick skillet and place over medium heat. Add the remaining butter to melt. Re-whisk the egg mixture. Taking a roll, place it in the egg mixture coating all sides allowing excess to drip off.
Place in the pan. Repeat for the remaining rolls. Cook for 3-4 minutes per side or until golden brown.
While they are on the last side being cooked, take the prosciutto and fold in half lengthways. Remove with tongs the savory French toast and place on top of the prosciutto. Gently roll the roll up grabbing the prosciutto wrapping it around.