Make the Raspberry Syrup:
Place the raspberries and water in a medium saucepan over medium-high heat. Using a potato masher or a fork, crush the berries. Bring the berries to a simmer, then lower the heat to medium-low and simmer for about 2 minutes.
Remove the pan from heat and pour the berries into a fine-mesh strainer set over a glass measuring cup. Using the back of a ladle or spoon, press on the berries to squeeze out as much juice as possible. Discard the seeds.
Return the raspberry juice to the pan, then add the sugar and lemon juice. Bring to a boil and simmer for about 4-5 minutes until the mixture reduces slightly. Remove from heat and chill in the refrigerator until completely cool.
Make the Filling:
In a stand mixer fitted with the whisk attachment, whip the heavy cream until stiff peaks form. Transfer the whipped cream to a separate bowl, then return the mixer bowl to the stand and replace the whisk attachment with the paddle attachment.
To the mixer, add the mascarpone cheese, sugar, ¾ cup of the raspberry syrup, and vanilla. Beat on high until combined. Add about a cup of the whipped cream and continue beating until incorporated. Gently fold in the rest of the whipped cream.
Assemble the Tiramisu:
One at a time, dip the ladyfingers into the raspberry syrup until soaked but not soggy, and arrange them in an even layer along the bottom of an 8-inch square dish.
Spread half of the mascarpone filling on top of the ladyfingers. Using a zester, microplane, or cheese grater, grate half of the white chocolate evenly on top. Place another layer of raspberry syrup-soaked ladyfingers on top of the filling.
Spread half of the remaining filling on top of the ladyfingers. Transfer the remaining filling to a pastry bag fitted with a large round tip (or a gallon ziplock bag with the corner cut off), and pipe the filling evenly over the tiramisu. Grate the rest of the white chocolate on top.
Cover with plastic wrap and refrigerate for at least 6 hours, preferably overnight, before serving.