Make the dough: To a stand mixer bowl, add the flour, sugar, salt and yeast. Using a dough hook, mix the dry ingredients to combine. While the mixer is still running, add in the egg, melted butter and vegetable oil. Slowly pour in the warm milk and continue to mix until a dough forms.
Once a dough forms, slightly increase the speed and knead for 5 minutes. At this point, the dough will be sticky. Flour a surface and use a wooden spoon to scrape the dough onto the floured surface. Sprinkle a teaspoon of plain flour onto the dough to make the outside less sticky then shape into a ball.
First rise: Add a couple drops of vegetable oil to a large bowl and spread the oil to cover the inside of the bowl. Place the dough inside the bowl and cover with a tea towel. Allow the dough to rise in a warm area for 2 hours or until the dough has doubled in size.
Fill the rolls: Whilst the dough is rising, you can prepare the filling. Add the unsalted butter, soft brown sugar and cinnamon to a bowl then mix to create a paste. Set aside until needed. Flour a surface and roll out the dough into a square shape which should be around ½ a centimetre or ¼ of a inch thick. Using a palette knife, smooth the cinnamon paste thinly onto the dough. Ensure it covers the whole surface.
Second rise: Tightly roll up the dough into a log shape then cut off both ends if they are not even. Cut the log into 9 even rolls then place them in a 9 x 9 inch square baking dish or tin. Ensure there is equal space between them all as they will rise and expand outwards. Cover them with a tea towel and allow to rise for 1 hour.
Bake the rolls: Preheat the oven to gas mark 4/350oF/180oC/160oC fan and bake for 28 - 30 minutes until lightly browned.
Make the icing: Combine the melted butter, icing sugar, milk and vanilla extract until you have a smooth paste then spread over the warm rolls.