Macerate the berries: In a medium bowl, combine the quartered strawberries, sugar, balsamic vinegar, and a tiny pinch of salt. Stir gently, then let sit at room temperature for 30 minutes. Stir once halfway through.
Toast the shortbread: While the berries macerate, preheat your oven to 350°F. Spread the crumbled shortbread pieces on a baking sheet in a single layer. Toast for 5-7 minutes, until they are just barely golden at the edges. Let them cool completely on the sheet.
Strain the berries (optional but smart): Pour the macerated strawberries into a fine mesh sieve set over a medium bowl. Press gently with a spatula to release most of the syrup. Set the syrup aside. You will use the syrup in the ice cream base and the solids as a layer.
Whip the cream: In the bowl of a stand mixer (or using a hand mixer), beat the heavy cream on medium-high speed until soft peaks form. This takes about 3-4 minutes. Stop when it looks like soft, billowy clouds.
Make the base: In a large separate bowl, whisk together the sweetened condensed milk, vanilla extract, lemon zest (if using), and the reserved strawberry syrup. Whisk until smooth and evenly pink.
Fold: Add about a third of the whipped cream to the condensed milk mixture and fold gently with a spatula until mostly combined. Add the remaining whipped cream and fold until no white streaks remain. The mixture should look like a uniform, pale pink mousse.
Layer it up: Spoon about a third of the ice cream base into your loaf pan. Scatter half of the reserved macerated strawberry solids and half of the toasted shortbread pieces over the top. Drizzle with a bit of extra syrup if you have it. Repeat with another third of the base, the remaining strawberries and shortbread, then finish with the final third of the base.
Swirl (optional but pretty): Take a butter knife and gently swirl it through the layers once or twice. You want ribbons of berry and pockets of shortbread, not a uniform mix.
Freeze: Cover the loaf pan with plastic wrap pressed directly onto the surface of the ice cream, then seal with foil or a lid. Freeze for at least 6 hours, preferably overnight. The wait is worth it.