Preheat your oven to 350 degrees Fahrenheit. Grease a 9×13-inch baking pan with butter. In a large bowl, mix the plums, sugar, flour, and vanilla and then transfer them to the baking dish.
Add the butter and work it into the flour with your fingertips or a pastry cutter until it is dry and crumbly. Add the milk and mix just until it forms a dough.
Flatten out small handfuls of the dough and arrange them on top of the plums. If you'd like, brush a little whisked egg white on top of the cobbler and sprinkle it with coarse sugar.