In two batches if necessary, blend together ice cream, cream cheese, milk, vanilla, lemon zest, and strawberries in a blender until smooth, 1 to 2 minutes.
Cooks' notes: If you want to use freshnot frozenberries, feel free to do so, but your milkshake won't be as thick. If you are lucky enough to have a bumper crop from your own garden or you bought loads of berries at the farmers' market, you can freeze them yourself. Wash and dry them and cut off the green tops. Spread the berries out on large rimmed baking sheets lined with towels, and freeze the berries uncovered, until firm, about a couple of hours, then transfer the frozen berries to heavy-duty resealable plastic bags.