To make the dough, place the flour, sugar, salt and yeast in the bowl of a stand mixer fitted with a dough hook and mix to combine.
Add the mashed banana, melted butter and most of the milk and mix to form a rough dough. Add the rest of the milk as needed to form a soft, slightly sticky dough.
Knead with the stand mixer on a high speed for about 5-10 minutes, until the dough is smooth and stretchy. The dough should be a bit sticky, but if it seems too wet you can add a little extra flour. If you don't have a stand mixer it can be kneaded by hand on a lightly oiled surface but it is quite a messy job.
Place the dough in an oiled bowl, cover and place in the fridge to rise overnight. (Or in a warm place until doubled in size - 1-2 hours.)
For the filling, mix together the light brown sugar and the cinnamon in a bowl and set aside.
Punch down the risen dough and give it a brief 30 second knead to knock out any air bubbles.
Roll the dough out on a floured surface to an approx 30x45 cm / 12x18 inch rectangle and spread the softened butter evenly over the surface.
Scatter over the cinnamon sugar mixture and press it down gently.
Roll the dough up tightly from one of the long edges into a sausage and trim off and discard the ends.
Use a sharp knife to slice the roll into 9 even slices then arrange them cut side down in a 23cm/9inch square tin lined with baking parchment.
Cover loosely and set aside to rise for 30-40 minutes until puffy. Meanwhile, preheat the oven to 180C/350F/gas mark 4.
Once the buns have risen, bake for 30-40 minutes, until the internal temperature reaches 94C/201F on a probe thermometer. Cover them loosely with tin foil partway through baking if
the tops start to colour too much.
Leave to cool in the tin for 30 minutes then carefully lift them out using the baking parchment and place on a wire rack.
To make the glaze, sift the icing sugar into a bowl and gradually stir in the milk and vanilla until it forms a thick but pourable glaze. Drizzle the glaze over the tops of the buns and serve. Store any leftovers in an airtight container for up to two days.