Preheat oven to 350F (or 175C).
Add egg whites to a mixing bowl. In another bowl, combine flour with sugar, and then add them to the egg whites. Whisk until well blended. Add vanilla extract and melted butter, whisk until the batter is smooth. The batter will look somewhat like sweetened condensed milk. Cover and refrigerate for about 20-30 minutes.
Spread almond slices into a thin layer on a baking sheet. Bake for about 5-6 minutes until lightly golden and fragrant. Set aside to let cool.
Remove the batter from the refrigerator, add almond slices and use a spatula to fold in.
Line a baking sheet with a silicone baking mat (or parchment paper). Add 1/2 tablespoon of batter onto the mat and use a knife (or back of a spoon, an offset spatula) to spread the batter into a thin circle about 3 - 3 1/2 inches (about 8 centimeters) in diameter. Make a few more circles, depending on the size of your baking sheet, spacing them about 2 inches apart. They will not change much in size.
Bake tuiles for 9-11 minutes until the edges are golden brown. Remove the baking sheet from the oven. Slide a knife underneath a tuile to lift it from the sheet, and then place it over a rolling pin (or a can, a bottle or in a tall glass) to form the curved shape. Work quickly to shape the remaining tuiles on the sheet.
You may need to bake several batches. Refrigerate batter while waiting to make a new batch. Before starting a new batch, let the baking sheet and mat cool completely before spreading batter onto it.