Warm the 1/2 cup milk to around 105–115°F (warm to the touch, not hot). Stir in the yeast and let sit about 5 minutes until foamy.

In a large bowl, combine 2 cups flour, 1/2 cup sugar, and 1/2 teaspoon salt. Stir in the yeast mixture, 1/4 cup softened butter, and 1/2 teaspoon vanilla extract until a shaggy dough forms.

Turn the dough onto a lightly floured surface and knead until smooth and elastic, about 6–8 minutes.

Place the dough in a greased bowl, cover, and let it rise in a warm spot until doubled, about 1 hour.

Preheat the oven to 350°F (175°C) and grease a 9x13-inch pan.

Punch down the dough and press it evenly into the prepared pan with your fingers.

Layer the sliced plums evenly over the dough, cut side down.

In a bowl, mix 1/2 cup brown sugar, 1/2 cup rolled oats, 1 teaspoon cinnamon, and the 1/2 cup cold cubed butter. Use a pastry cutter or your fingers to rub the butter in until the mixture is crumbly. Sprinkle evenly over the plums.

Bake for 35–40 minutes until the topping is golden and the edges are bubbling.

Let the cake cool slightly and serve warm.