In a medium bowl, stir together strawberries, lime juice, and 1/2 cup sugar. Let stand at room temperature 2 hours. Transfer mixture to food processor or blender; puree until smooth.

Place milk and remaining 1 cup sugar in mixing bowl; mix on medium-low speed 2 minutes or until sugar is dissolved. Add heavy cream, vanilla and pureed strawberries; mix on low 1 minute or until well combined. Pour mixture into your ice cream machine and freeze according to instructions for your machine (mine took about 25 minutes).

Meanwhile, in small bowl, melt butter in microwave. Add graham cracker crumbs and stir until crumbs become clumpy and lumpy (this is a good thing!).

When ice cream has finished churning (mixture will be soft), begin dividing ice cream between two 1-quart plastic containers. Add one scoop of ice cream to each container, then sprinkle with some graham cracker crumbs and sprinkles. Repeat layers 5 times. Transfer ice cream to freezer to harden completely (at least 4 hours).