Prepare the batter by mixing the warm milk and vanilla together. Stir in the butter and continue to mix until the butter is melted and well combined.
In a large bowl, combine the flour, yeast, sugar, salt and egg and mix well. Slowly add the water and milk mixture, while stirring, and continue to mix until well combined. Once the batter starts to come together be sure to scoop the sides of the bowl while continuing to mix, a rubber spatula works well. The dough is ready when it is soft, yet a little sticky, and pulls away from the bowl easily. Add more flour if necessary.
Transfer the dough to a lightly floured work surface and knead for 5 minutes, adding a little more flour if it's still too sticky. Cover with a tea towel and let the dough rest for 10 minutes.
To make the filling: Mix the melted butter, brown sugar and cinnamon together until well incorporated. Set aside.
Dust a rolling pin with flour. Roll the dough out into a large rectangle about 12 x 9-in, or approximate. Spread the filling on the dough, using the back of a spoon or rubber spatula is helpful. Try to spread it out as evenly as possible.
Tightly roll up the dough lengthwise, press and pinch the seam to make certain it is secure. With a sharp knife, make slices about 1-in apart or so. Depending on the sizes cut, the yield will vary. Place each roll, cut side up, into a greased pie pan or a large skillet lined with parchment paper. It’s fine for the rolls to touch a little, but you’ll want to keep some room for them to rise and expand while baking. Use a tea towel and cover. Let the rolls rise for about 30 minutes.
In the meantime, preheat the oven to 375 F (191 C.). Bake the rolls, uncovered, for 20-25 minutes or until golden brown and cooked through. Let the rolls rest for 5 minutes before glazing.
To make the glaze: Mix the cream cheese, butter, confectioner’s sugar and vanilla together. Using an electric beater or stand mixer works best to incorporate any lumps of butter and to yield a smooth icing. Use milk or dark coffee to thin the glaze, only if necessary. Noting that coffee lends a distinct flavor and taupe color to the outcome. Spread the glaze on the buns while warm, noting that you might not use it all. Serve the rolls warm and enjoy!