Let one container of the ice cream soften at room temperature for about 30 minutes. Mix the raspberries, lemon juice and sugar in a medium pot. Let sit until the berries start to release their juices, 10 minutes, and then turn the heat to medium. Bring to a simmer, stirring occasionally, and then continue to cook until the raspberries have lost their shape and turned into a thick sauce, 16 to 18 minutes. Remove from the heat and cool.
Press the berry mixture through a strainer to remove the seeds and set this sauce aside. You should have about 1/4 cup. Put half of the container of softened ice cream in a medium bowl (metal is best as it will re-chill faster). Drop half the raspberry sauce over the ice cream in spoonfuls and then scatter with half of the chopped chocolate. Stir just a few times to mix in the ingredients. Some of the ice cream will turn pink but there should still be swirls of raspberry. Add the rest of the container of ice cream, raspberry sauce and chocolate. Repeat the mixing process. Press a piece of plastic wrap on the surface of the ice cream and return to the freezer until no longer melty but still workable, 30 to 45 minutes. About halfway through cooling down the ice cream, pull the second container of ice cream from the freezer to soften.
Add the second container of vanilla ice cream, in big scoops, into the bowl of frozen ice cream. Mix one or two times with a wooden spoon or spatula, recover and return to the freezer until hardened, at least 2 hours or overnight. Don't worry if the ice cream doesn't look evenly mixed, when you go to scoop it, you will get the perfect blend of all the flavors.
Adapted from Bo's Express Ice Cream from the Babcock, Hall Dairy Store for Cooking Channel