Add the raspberries and water to a saucepan over medium heat. Bring the mixture to a gentle boil.
Crush the softened berries with a potato masher or fork. Reduce the heat and simmer for approximately 10 minutes.
Remove the saucepan from the heat. Pour the raspberry mixture through a fine-mesh strainer, pressing gently to extract the liquid. Discard the remaining seeds and pulp.
Return the strained raspberry liquid to the saucepan over low heat. Add the granulated sugar and stir until completely dissolved.
Stir in the fresh lemon juice and simmer gently for another 10 minutes.
Strain the raspberry sauce once more for an especially smooth consistency. Transfer it to a wide bowl, cover, and refrigerate until completely chilled.
To prepare the mascarpone filling, add the cold heavy cream to the bowl of a stand mixer fitted with the whisk attachment. Whip on medium-high speed until stiff peaks form. You can also use an electric hand mixer.
Transfer the whipped cream to a separate bowl.
In a clean mixing bowl, combine the mascarpone, granulated sugar, alcohol-free vanilla, and 3/4 cup chilled raspberry sauce. Beat just until smooth and combined.
Add approximately 1 cup of the whipped cream and mix gently until incorporated.
Add the remaining whipped cream and fold it into the mascarpone mixture with a spatula. Fold gently to preserve as much air as possible.
Pour some of the remaining chilled raspberry sauce into a shallow bowl. Briefly dip the ladyfingers into the sauce until coated and slightly softened. Avoid soaking them until they begin falling apart.
Arrange a single layer of raspberry-dipped ladyfingers across the bottom of an 8×8-inch or 9×9-inch dish. Trim the cookies when necessary to fit.
Spread half of the mascarpone mixture evenly over the ladyfingers.
Sprinkle half of the finely grated white chocolate over the mascarpone layer.
Add another layer of raspberry-dipped ladyfingers, followed by the remaining mascarpone filling.
Sprinkle the remaining grated white chocolate evenly over the White Chocolate Raspberry Tiramisu.
Cover the dish and refrigerate for at least 6 hours. For the best texture and cleanest slices, refrigerate overnight.
Serve chilled with fresh raspberries and additional raspberry sauce, if desired.