Preheat your oven to 300°F (150°C). Line a baking sheet with parchment paper.

Place the halved strawberries in a single layer on the prepared sheet. Sprinkle with 1/4 cup sugar and drizzle with lemon juice. Toss gently to coat.

Roast for 30 to 35 minutes, stirring once halfway through, until soft and syrupy. Let cool slightly, then blend into a smooth purée. Refrigerate until cold.

In a medium saucepan over medium heat, combine the milk, softened cream cheese, and corn syrup. Whisk until smooth and just warm, about 3 minutes. Do not boil.

In a separate bowl, whisk the egg yolks with 2/3 cup sugar until pale and thick.

Slowly add a bit of the warm milk mixture into the yolks while whisking constantly to temper them.

Pour the tempered yolk mixture back into the saucepan with the remaining milk mixture. Cook over medium heat, stirring often, until thick enough to coat the back of a spoon (170°F/77°C), about 5–7 minutes.

Remove from heat. Stir in salt, vanilla, and heavy cream. Strain through a fine-mesh sieve into a large bowl to remove any solids. Cover and chill in the fridge at least 4 hours or overnight.

Once the base is cold, mix in the strawberry purée.

Churn in an ice cream maker according to the manufacturer’s instructions, usually about 20–25 minutes, until it reaches soft-serve consistency.

Layer the churned ice cream into a freezer-safe container, sprinkling in crushed graham crackers between scoops.

Press plastic wrap or parchment paper directly onto the surface and freeze until firm, at least 4 hours.