Weigh the empty 2-quart saucepan, then add the strawberries, 120 grams granulated sugar, and lemon juice. Cook over medium heat, stirring and crushing the berries, until bubbling steadily and reduced to 260 grams after subtracting the pan weight, about 12 to 16 minutes. The mixture should look glossy and leave a brief trail when stirred. Blend carefully with an immersion blender until mostly smooth, leaving fine fruit flecks, then transfer the puree to a heatproof bowl.
Whisk the remaining 70 grams sugar, milk powder, and 10 grams cornstarch in the cleaned saucepan. Gradually whisk in the milk and cream, then cook over medium heat, stirring constantly, until the base reaches 82°C and lightly coats a spatula, about 5 to 8 minutes. Whisk the hot base gradually into the softened cream cheese until smooth. Add the strawberry puree, vanilla, and salt, then chill over an ice bath. Cover and refrigerate at least 4 hours, until thoroughly cold.
Line a 23 by 33 cm metal pan with parchment, leaving enough overhang to lift the filling later. Churn the cold strawberry base according to the ice cream maker instructions, usually 20 to 30 minutes, until it holds soft ridges and resembles thick soft-serve ice cream. Immediately spread it evenly in the lined pan to approximately 1.5 cm thick, smoothing the surface with an offset spatula. Cover the surface and freeze on a level shelf for at least 4 hours, until firm enough to cut cleanly.
Whisk the flour, 18 grams cornstarch, baking soda, and cookie salt in a medium bowl. In a stand mixer fitted with the paddle, beat the butter, brown sugar, and 75 grams granulated sugar on medium speed for 2 minutes, until creamy but not excessively fluffy. Scrape the bowl, add the egg, yolk, and vanilla, and mix until emulsified. Add the dry ingredients on low speed and stop as soon as no dry flour remains; prolonged mixing would make the frozen cookies tougher.
Divide the dough into 24 portions of about 35 grams each. Roll each portion smoothly, arrange on parchment-lined baking sheets with at least 5 cm between them, and flatten to discs about 6 cm wide. Refrigerate for 30 minutes so the butter firms and the cookies spread evenly rather than becoming thin and brittle. During the final 20 minutes, position a rack in the center of the oven and preheat to 170°C. Keep the second tray chilled while the first tray bakes.
Bake one tray at a time for 9 to 11 minutes, rotating after 7 minutes. The edges should be set and pale golden, while the centers should look puffed, matte, and soft without appearing wet or glossy. Let the cookies rest on the hot baking sheet for 8 minutes, where residual heat finishes the centers safely without drying them. Transfer carefully to a wire rack and cool completely, about 30 minutes. Pair cookies of similar diameter before assembling the sandwiches.
Chill a parchment-lined tray in the freezer for 10 minutes. Lift the firm ice cream slab from its pan and use a 7 cm round cutter to stamp 12 discs, dipping and drying the cutter between portions if needed. Place one cookie flat side up, add a strawberry ice cream disc, and cover with a second cookie flat side down. Press only until the filling meets the edges. Work with two or three sandwiches at a time, returning the filling to the freezer whenever it begins to soften.
Arrange the completed sandwiches on the chilled tray and freeze uncovered for 45 to 60 minutes, until the cookies and filling feel firmly joined. Wrap each sandwich snugly in parchment, then place it in foil or an airtight freezer container to block air and freezer odors. Freeze until serving. For the most comfortable bite, unwrap and rest each sandwich at room temperature for 5 to 8 minutes. The filling should yield slightly when pressed while the cookies remain cold and flexible.
