Combine raspberries, sugar, water, and lemon juice in a saucepan over medium/low heat. Stir frequently until berries release their juices.

Increase heat to medium and simmer while stirring constantly until berries break down and sauce thickens slightly.

Remove from heat and strain through a fine mesh strainer, adding 2 Tablespoons of seeds back to the sauce.

Allow sauce to cool while preparing the crust and cheesecake.

Preheat oven to 325F (165C).

In a bowl, stir together graham cracker crumbs, granulated sugar, and brown sugar. Add melted butter until all crumbs are moistened.

Press mixture evenly into the bottom and sides of a 9" springform pan.

In a mixing bowl, combine cream cheese and sugar using an electric mixer until smooth and lump-free.

Add sour cream and vanilla extract, stirring on low speed until combined.

Lightly beat an egg in a small bowl and add to batter, stirring on low speed until just combined. Repeat with remaining eggs.

Pour half of the batter over the crust, then add ⅓ cup of the raspberry sauce on top.

Dollop remaining batter over the sauce and smooth. Drizzle with another ⅓ cup of raspberry sauce and swirl with a toothpick.

Bake in preheated oven for 45-55 minutes, until center is jiggly but mostly set.

Turn off oven and crack the door open. Let cheesecake cool in the oven for 1 hour.

Cool at room temperature for an hour, then refrigerate for at least 6 hours, preferably overnight.

Serve topped with reserved raspberry sauce.