In a small bowl, combine warm water and yeast. Let stand for 5-10 minutes until foamy.
In a large bowl, whisk together warm milk, sugar, and salt. Add the yeast mixture and lightly beaten egg. Mix well.
Gradually add 3.5 cups of flour to the wet ingredients, mixing until a soft dough forms. Turn out onto a lightly floured surface and knead for 8-10 minutes, adding the remaining flour as needed, until the dough is smooth and elastic. Incorporate the softened butter during the last few minutes of kneading until fully combined.
Place the dough in a greased bowl, cover with plastic wrap, and let rise in a warm place for 1-1.5 hours, or until doubled in size.
Punch down the dough and turn it out onto a lightly floured surface. Roll the dough into a 12x18-inch rectangle.
In a small bowl, combine brown sugar and cinnamon for the filling. Brush the rolled-out dough with melted butter. Sprinkle the sugar and cinnamon mixture evenly over the dough.
Starting from the long side, tightly roll up the dough. Cut the roll into 12 equal slices.
Place the honey bun slices into a greased 9x13-inch baking pan, cut-side up.
Cover the pan and let the buns rise for another 30 minutes.
Preheat oven to 375°F (190°C).
Bake for 20-25 minutes, or until golden brown.
While the buns are baking, prepare the glaze: In a small bowl, whisk together honey, melted butter, and vanilla extract.
Remove the buns from the oven and immediately drizzle the glaze over the hot buns.
Let cool slightly before serving.
Notes
For extra gooeyness, you can add a sprinkle of chopped pecans or walnuts on top of the glaze while it's still warm.