Place the strawberries into a food processor and pulse until completely pureed.
Transfer pureed strawberries to a medium saucepan and add the sugar. Cook over medium heat until mixture begins to simmer.
Pour mixture through a fine mesh sieve, using a rubber spatula to push as much juice from the berry solids as possible. You should end up with about 3/4 - 1 cup berry juice. Transfer berry juice to a glass measuring cup and chill in the refrigerator until cooled completely.
In the chilled bowl of a stand mixer fitted with the whisk attachment, whip the heavy cream until stiff peaks form.
Pour sweetened condensed milk into a large mixing bowl. Stir in vanilla. Gently fold a third of the whipped cream into the condensed milk until combined. Add the rest, and continue folding until you see just a few streaks of white.
Remove half of the mixture, scooping it into another bowl. To this half of the mixture, fold in the cooled berry juice until just combined.
Drop a few spoonfuls of the berry custard and vanilla custard into a freezer-safe container. Sprinkle with some of the chopped shortbread cookies. Repeat this layering process 2-3 more times until you've used up both custards and all of the cookies. Using a skewer or butter knife, swirl figure-eights throughout the ice cream.
Chill the ice cream in the freezer for at least 6 hours, preferably overnight.
Scoop, serve, and enjoy!