To make the topping: In a small bowl, combine the brown sugar, butter, and cinnamon. Use a flexible spatula to mix slowly to combine, then mix vigorously until smooth and slightly fluffy. Set aside while you prepare the batter.
To make the cake: In a medium bowl, whisk the flour, sugar, baking powder, and salt until combined.
Add the yogurt, melted butter, egg, and vanilla and continue whisking until the batter is mostly smooth; some small lumps are OK.
Transfer the batter to the prepared pan and spread it into an even layer.
Dollop heaping teaspoons of the topping all over the top of the batter (a scant teaspoon scoop works well here), then use a toothpick or a butter knife to swirl the topping into the batter in a decorative pattern. If desired, reserve 1 or 2 teaspoons of the topping to swirl into the frosting when finishing the cake.
Bake the Cinnamon Roll Cake for 30 to 35 minutes, until a toothpick or thin knife inserted into the center of the cake comes out clean.
Remove the cake from the oven and let it cool in the pan on a wire rack for about 20 minutes. Transfer the cake to a wire rack to continue cooling while you make the frosting.
To make the frosting: In a medium bowl, combine the confectioners’ sugar and cream cheese. Use a flexible spatula to mix slowly to combine, then vigorously until smooth.
Gently transfer the cake to a serving plate. Spoon the frosting over the top of the cake (it’s OK if it’s still slightly warm) and use the back of the spoon or a small offset spatula to spread it evenly. To create a swirled look reminiscent of a cinnamon roll, drop small spoonfuls of the reserved topping over the frosting and lightly swirl to incorporate. Serve the cake still slightly warm or at room temperature.
Storage information: Store leftover Cinnamon Roll Cake, covered, at room temperature for several days; freezing is not recommended.
