Place the tapioca pearls in a container large enough to accommodate the soaking water as well; add enough water to cover the tapioca by about 3 inches. Refrigerate and soak overnight.
Drain the tapioca, discarding the water.
Whisk the tapioca together with the milk, salt, sugar and egg in a large saucepan. Place the pot over medium heat and cook for about 15 minutes, whisking constantly, to form a thickened pudding. Remove the pan from the heat and let sit for 15 minutes. Stir in the vanilla and almond extracts.
Transfer to a large bowl; cover and refrigerate for about 2 hours, until fully set and chilled.
For the filling:
Combine the peaches, sugar and 1/4 cup of the water in a large saucepan over medium heat. Cook for 8 to 10 minutes, stirring, until the fruit has softened to your liking.
Mix together the remaining 2 Tbsp of water and the cornstarch, stirring until the cornstarch has dissolved. Add to the fruit and cook for 2 to 3 minutes or until the mixture thickens. Remove from the heat and stir in the vanilla and almond extracts. Taste, and add sugar as needed. Refrigerate until well chilled.
Line up small dessert cups or bowls. Distribute half of the tapioca pudding evenly among the bowls, then use about half of the peach filling to create a second layer in each of the portions. Use the remaining tapioca for a third layer, then top with the filling. Garnish with sliced almonds and serve immediately.