Place white chocolate and ¼ cup milk in a double broiler over medium-heat (or in a heat-proof bowl set over a pot of lightly boiling water) and gently stir until melted. Once chocolate is melted, set bowl aside and let cool a bit.
In a large bowl, whisk together eggs and ¾ C sugar, until light and fluffy. Whisk in cream, remaining cup of whole milk, and vanilla.
Stir cooled white chocolate into the mixture. Cover bowl with plastic, touching plastic wrap to the top of the mixture, and place in fridge to chill, at least 4 hours.
Once you're ready to process ice cream, place sliced strawberries in a bowl, along with remaining 2 Tbsp sugar, and mash together a bit. Let sit for at least 5 minutes.
Using your ice cream maker, process ice cream mixture according to manufacturer’s directions. Once the ice cream is starting to thicken, about 5 minutes before it's finished processing, add in sliced strawberries.
Put ice cream into containers and place in fridge to chill for at least 6 hours.