Lightly grease a fluted tube pan (such as Bundt) with vegetable oil; set aside.
Make dough: Combine warm water, yeast, and 1 teaspoon sugar in a large bowl. Let stand until bubbly and foamy, about 5 minutes.
Stir in 1 cup flour, warm milk, egg, 3 tablespoons sugar, butter, salt, and vanilla extract until combined. Add remaining 2 1/2 cups flour. Mix and knead until dough comes together in a ball. Cover with plastic wrap and let rise in a warm place for 30 minutes.
Turn dough out onto a lightly floured surface and punch down gently to deflate. Divide dough into small or medium balls.
Make coating: Mix together brown sugar and cinnamon in a small bowl. Dip each dough ball in melted butter, remove with a fork, then roll in cinnamon sugar mixture until well coated.
Arrange coated dough balls in the prepared pan. Cover with plastic wrap or a damp cloth and let rise until almost doubled in size, about 45 minutes.
Preheat the oven to 350 degrees F (175 degrees C).
Bake in the preheated oven until golden brown, about 35 minutes. Invert onto a large plate; cool for 10 minutes before serving.
