In a bowl, combine warm water, warm milk, and yeast. Let sit until frothy, about 10 minutes.

Stir in sugar, melted butter, eggs, and salt. Gradually add flour and mix until a soft dough forms. Knead 6–8 minutes by hand or with a stand mixer.

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

Meanwhile, in a saucepan over medium heat, melt butter, brown sugar, and vanilla until bubbly. Pour evenly into the bottom of a greased 9×13-inch pan.

Punch down dough and roll into a 12×18-inch rectangle on a floured surface. Spread softened butter, sprinkle brown sugar and cinnamon mixture evenly.

Roll dough tightly from the long side, then cut into 12 equal pieces. Arrange cut sides up in the prepared pan. Cover and let rise 30–45 minutes.

Preheat oven to 350°F (175°C). Bake buns 25–30 minutes or until golden brown and caramel is bubbling.

Let cool 5 minutes, then invert the pan onto a serving platter so the caramel glaze coats each bun.