Stem and roughly slice the strawberries (you'll puree them later, so don't worry about being too neat).
Add lemon juice, zest, and 1/3 cup sugar to the strawberries, stir gently.
Cover and store in the fridge for about two hours.
After about an hour and a half, combine the milk and the rest of the sugar until the sugar dissolves (about 2 minutes with a hand-held mixer on low).
Stir in the heavy cream, and some juice from the strawberries (but NOT the berries!).
Add the mix to the ice cream maker and freeze per manufacturer's instructions.
About 10 minutes before the ice cream is finished, puree most of the berries. Reserve a few berries as sliced for garnish or combine with the pureed berries in the ice cream maker for chunkier ice cream.
Add the strawberries to the ice cream maker for the last 5 minutes of freezing.
Notes: My ice cream maker doesn't get the ice cream as firm as I like it, so at this point, I remove it from the ice cream maker and put in in the freezer for about 4 more hours. But that's a personal choice. Also, this ice cream is pale pink, so if you want it to be pinker, add a drop or two of red food coloring.