Preheat your oven to 350ºF (180ºC) and adjust the oven rack to the middle position. Line a 9-inch (23cm) square baking pan with parchment paper, leaving some overhang for easy removal later.

In a large bowl, combine the sugar, melted butter, oil, eggs, yoghurt, milk, and vanilla. Whisk until the mixture is smooth and well blended.

In another bowl, mix together the gluten free flour blend, baking powder, xanthan gum, and salt. Gradually add the dry ingredients to the wet mixture, whisking until you achieve a smooth, slightly loose batter without any flour lumps.

Pour the cake batter into the prepared baking pan, smoothing it out into an even layer.

In a separate bowl, combine the light brown sugar, cinnamon, and gluten free flour blend. Stir in the melted butter until you have a loose, paste-like mixture.

Drop small dollops of the cinnamon-sugar filling over the cake batter, using about 1-2 teaspoons for each dollop. Gently swirl the filling into the batter with a skewer or the end of a knife.

Bake the cake for 25-28 minutes, or until the top is golden brown and a toothpick inserted into the center comes out clean.

Let the cake cool in the pan for about 20 minutes while you prepare the icing.

In a bowl, mix the cream cheese, powdered sugar, and vanilla until smooth and free of lumps. The mixture will thicken initially but will loosen as the sugar absorbs moisture from the cream cheese.

Add the milk and mix until the icing is smooth. It should be runny but not overly thin; if it's too thick, add 1-2 teaspoons of extra milk to achieve the desired consistency.

Pour the icing over the warm cake, using the back of a spoon to spread it evenly into all the crevices. You can choose to use less icing if you prefer a less sweet dessert.

Slice the cake and serve. It can be enjoyed warm or at room temperature. The gluten free cinnamon roll cake is best on the day it is baked but can be stored in an airtight container in a cool, dry place for 2-3 days.