1. Have ready an 8- by 11-inch baking dish.
  2. In a large mixing bowl, beat mascarpone with an electric mixer. Add the condensed milk and beat until combined. Add the peanut butter and vanilla extract; beat again. Set aside.
  3. In a separate mixing bowl with clean beaters, beat heavy cream until soft peaks form. Gently fold into the peanut butter mixture. Set aside.
  4. Place the milk in a wide shallow bowl or salad plate. Dip 12-15 of the lady finger cookies into the milk and lay evenly across the bottom of the baking dish. Break them into smaller pieces as needed so they fit into all the corners and spaces. There will be some gaps between the cookies.
  5. Spread one third of the peanut butter mousse over the cookies. If desired, sprinkle with some of the chocolate shavings. Dip 12-15 more lady finger cookies into the milk and create another layer over the mousse. Top with the next third of the mousse, making sure to spread it evenly to the edges. Sprinkle again with chocolate, if using.
  6. Repeat the layering with the remaining lady fingers and mousse. Top with more chocolate shavings, if desired. Cover and place in fridge for at least 3-4 hours, or preferably overnight. The cookies will continue to absorb liquid and become soft, and the mousse will set up.
  7. To serve, cut the pave into 12-16 pieces. Best served chilled.

Note: This can be made 3-4 days in advance. It gets better as it sits. Will keep for about a week in the fridge.