Line a small tray or quarter sheet pan with parchment paper. In a medium bowl, beat the cold cream cheese on medium-high speed until completely smooth, about 1 minute. Stir in the melted, cooled white chocolate until glossy and unified.
Spoon or pipe 20 rounded 2-teaspoon mounds of the cheesecake mixture onto the lined tray. Freeze until firm to the touch, 30–40 minutes, and keep frozen until assembly.
For the raspberry swirl, place the raspberries and 1/3 cup sugar in a saucepan. Lightly mash and let stand 10 minutes to draw out the juices. Set over medium heat and simmer, stirring often, until syrupy and slightly thickened, 8–10 minutes.
While warm, press the raspberry mixture through a fine-mesh sieve into a bowl to remove the seeds (work it through to extract as much liquid as possible). You should have just under 1/3 cup. Refrigerate to cool while you make the dough.
Preheat the oven to 360°F. Line two baking sheets with parchment paper.
In a medium bowl, whisk together the flour, baking powder, baking soda, and salt.
In a large bowl, beat the softened butter, granulated sugar, and lemon zest on high speed until very light and fluffy, 2–3 minutes. Add the egg and vanilla; beat on medium-high until pale and airy, about 1 minute more.
Add the dry ingredients and mix on low just until a soft dough forms; do not overmix.
Marble the dough: Spread the dough in an even layer in the bowl, then drizzle the cooled raspberry syrup over the surface in 3–4 ribbons. Using a spatula, fold the dough over itself 3–4 times so you see streaks of raspberry throughout without fully blending it in.
Assemble: Scoop the dough into 20 portions with a 2-tablespoon scoop. Working one at a time, press a portion into a shallow 2-inch cup in your palm. Nestle a frozen cheesecake mound in the center, pinch the dough to seal completely, then place seam-side down and gently press to about 1/2 inch thick. Repeat, keeping the cheesecake centers in the freezer until needed.
Sprinkle the tops generously with turbinado sugar. Arrange 6 cookies per prepared sheet and bake for 10–12 minutes, until the edges are set and lightly golden while the centers remain slightly puffy. If desired, use a large round cutter to gently scoot the edges into a perfect circle right out of the oven.
Cool on the baking sheet for 10 minutes, then transfer to a rack to cool completely before serving.