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Heat the oven to 325 degrees. Lightly spray a 9-by-13-inch pan with nonstick spray, and line with parchment paper, leaving about 2 inches of excess on each side to eventually help you lift the cooked cake out of the pan.
In a medium bowl, whisk together the flour, cinnamon, baking powder and salt.
In a small pot, melt the butter over medium heat. Continue to heat the butter, stirring occasionally, until the solids begin to brown and the butter smells nutty, 9 to 10 minutes. Transfer to a medium bowl and cool for 20 minutes.
Add the brown sugar and granulated sugar to the bowl and whisk to combine. Add the eggs one at a time, whisking well to fully incorporate each before adding the next. Add the vanilla and whisk to combine.
Add about half of the flour mixture and whisk to incorporate. (If necessary, switch to a spatula or wooden spoon at this point.) Add the sour cream and mix to combine. Add the remaining flour and mix to combine.
Transfer the batter to the prepared pan and spread into an even layer. Press the 12 apple segments partly into the batter, spacing them evenly throughout and keeping in mind that you want each of 12 square slices to have one apple quarter.
Transfer to the oven and bake for 60 to 70 minutes. A toothpick inserted into the center of the cake should come out clean, and you should be able to easily pierce an apple with a small knife. Let the cake cool completely in the pan.
Just before serving, use the parchment flaps to lift the cake out of the pan. Combine the confectioners’ sugar with the cinnamon and dust the surface of the cake generously with the cinnamon sugar. Cut into 12 square pieces or 24 triangles.