In a large mixing bowl, beat 1 cup butter and granulated sugar together on medium speed until very light and fluffy, about 3 minutes.
Add the eggs one at a time, beating well after each addition and scraping down the sides of the bowl so everything is well combined.
Mix in the vanilla bean paste on low speed until evenly distributed.
In a separate bowl, whisk together the flour, cornstarch, and kosher salt.
Add the dry ingredients to the butter mixture in two additions, mixing on low speed just until a soft dough forms and no dry streaks remain.
If the dough is very soft or sticky, chill it in the refrigerator for 15 to 20 minutes to firm up slightly.
Preheat the oven to 350°F. Line two baking sheets with parchment paper.
Scoop tablespoon sized portions of dough, roll into balls, and place them on the prepared baking sheets, leaving a little space between each.
Gently flatten each ball slightly with your fingertips or the bottom of a measuring cup to create even rounds.
Bake one sheet at a time for 10 minutes until the centers look set and the edges are just barely starting to turn golden.
Let the cookies cool on the baking sheet for 5 minutes, then transfer to a wire rack to cool completely.
While the cookies cool, place the strawberries in a small saucepan and cook over medium heat, stirring occasionally, until they break down and the mixture thickens and reduces, 8 to 10 minutes.
Transfer the cooked strawberries to a blender or food processor and blend until smooth, then let the puree cool completely to room temperature.
In a clean mixing bowl, beat 1/2 cup butter on medium speed until smooth and creamy.
Gradually add the confectioners sugar, mixing on low at first, then increasing to medium until thick and smooth.
Beat in the cooled strawberry puree and heavy cream until the buttercream is fluffy and pipeable, adding more confectioners sugar if needed to thicken.
Once the cookies are completely cool, flip half of them over so the flat sides face up.
Spoon or pipe a swirl of strawberry buttercream onto the flat side of each flipped cookie, leaving a small border around the edge.
Top each with a second cookie, flat side down, and gently press until the filling spreads to the edges to form sandwich cookies.
Chill the assembled cookies for 20 minutes to help the filling set, then serve slightly cool or at room temperature.