In a medium saucepan over medium heat, combine sliced peaches, water, and granulated sugar, stirring gently. Once the mixture boils, reduce heat and let simmer for about 15 minutes until slightly thickened. Strain through a fine mesh sieve and let cool completely.

Whip heavy cream on high until stiff peaks form in a mixing bowl. In another bowl, mix mascarpone cheese, remaining sugar, chilled peach syrup, vanilla extract, and almond extract until smooth. Fold whipped cream into the mascarpone mixture.

Dip each savoiardi ladyfinger briefly into the chilled peach syrup and arrange in an 8-inch dish. Spread half of the mascarpone filling over the ladyfingers. Repeat with another layer of dipped ladyfingers and remaining filling.

Cover the dish with plastic wrap and refrigerate for at least 6 hours, preferably overnight. Serve chilled, garnished with peach compote or a drizzle of syrup.