Make the cinnamon rolls (skip steps 1-7 if you use store-bought buns).
Prepare the dough by whisking flour, sugar, and salt in a large mixing bowl and setting it aside.
In a deep pan, combine milk and butter, then heat on stove until the butter melts and the mixture is warm but not hot (104°F/40°C). Stir in the yeast until it fully dissolves.
Introduce this liquid mixture to the set-aside dry ingredients, adding an egg as well. Combine using a spatula or a stand mixer with a paddle attachment on medium speed until a soft, pliable dough is achieved.
Turn the dough onto a floured surface and knead with floured hands for around 5 minutes until it forms a smooth ball. Add a bit more flour if the dough feels overly sticky or soft.
Place the dough in a bowl that’s been lightly greased, covering it loosely. Let it sit for approximately 10 minutes while you prepare the filling.
For the filling, roll the rested dough into a rectangle measuring approximately 12 x 8 inches (30x20 cm). Spread softened butter over the surface, followed by a mixture of cinnamon and brown sugar sprinkled evenly on top.
Carefully roll the dough to form a log, slicing it into 8 equal portions. Arrange these rolls in a greased round baking pan.
Allow the rolls to rise by covering the pan with a clean towel, placing it in a warm area for 60 minutes until they have doubled in size.
Heat the oven to 180°C (356°F).
In a bowl, mix eggs, milk, vanilla and cinnamon until you get a homogeneous composition.
Pour the egg mixture over the rolls, distributing evenly. Then sprinkle with maple syrup.
Put the cinnamon roll casserole in the hot oven and leave it to bake for 30 minutes or until it is well browned on the surface.
Mix the ingredients for the glaze in a bowl.
Serve the cinnamon bun casserole hot and drizzled with glaze.