In a large mixing bowl, use a handheld mixer to combine the sugars and lemon zest until fragrant. Add the room-temperature butter and beat on medium speed until light and fluffy. Reduce the speed to low and mix in the milk and vanilla extract.

Add the oats, flour, cornstarch, baking soda, and salt. Mix on low speed just until a soft dough forms. Gently fold in the frozen raspberries, being careful not to overmix or break them apart. Cover the dough and refrigerate for 15 minutes.

Preheat the oven to 350°F and line a large baking sheet with parchment paper. Use a ~1½-tablespoon cookie scoop to portion the dough onto the prepared baking sheet, leaving about 2 inches between each cookie. Bake for 14 minutes, or until the edges are lightly golden and crisp. The centers may look slightly underbaked, which is okay - they will continue to set as they cool. Let the cookies cool on the baking sheet for 10 minutes, then transfer to a wire rack to cool completely.

While the cookies cool, add the cream cheese and softened butter to a large mixing bowl. Using a handheld mixer, beat until smooth and creamy. Gradually add the powdered sugar and mix on low speed until incorporated. Add the lemon juice, lemon zest, vanilla extract, and salt, then beat until smooth and fluffy.

In a separate bowl, use a handheld mixer to whip the heavy cream to stiff peaks. Gently fold the whipped cream into the cream cheese mixture until smooth and fluffy.

Once the cookies are completely cooled, pipe or spread the lemon cream filling onto the flat side of half of the cookies. Top with the remaining cookies and gently press together.