Place strawberries and 2 tablespoons of granulated sugar in a small saucepan over medium heat. Cook, stirring often, until blackberries are falling apart - about 10 minutes. Remove from heat and let cool.
Place butter and remaining granulated sugar in the bowl of a stand mixer fitted with a paddle attachment, and beat on medium-high speed until smooth and well creamed, scraping down the bowl of the mixer as needed. Add strawberry mixture, and fold until combined.
Next, add egg and vanilla, mixing until combined. Meanwhile, in a medium bowl, combine gluten-free all purpose flour, salt, and baking soda. Add the flour mixture to the butter mixture, and mix on low until just combined, scraping the bowl as needed.
Line a baking sheet with parchment paper, and scoop cookies onto parchment paper, using a 1 tablespoon-sized ice cream scoop. Place the cookies in the refrigerator to chill for at least 1 hour. While the dough chills, make the cheesecake mixture by whisking together the cream cheese, granulated sugar, vanilla extract, and salt in a medium bowl until well combined. Set aside, keeping at room temperature.
Preheat the oven to 375°F. Remove the cookie dough balls from the refrigerator and place on a parchment lined baking sheet, about 2 inches apart. Gently press your thumb into the center of each cookie dough ball, pressing outward, until a 1 - inch crater is formed.
Once each cookie has a crater formed in the center, dollop about 2 teaspoons of cheesecake mixture into the center of each cookie. You can do this by using a small teaspoon, a small scoop or a piping bag.
Place the cookies in the oven and bake for 6-8 minutes, rotating halfway through baking. Once the cookies are baked, let cool on the baking sheet for 5 minutes before moving to a wire cooling rack to cool completely.
Store in an airtight container for up to 3 days.