Add butter to a skillet over medium heat. Let it melt and cook, stirring occasionally, until it smells nutty and turns golden brown (about 5–7 minutes). Remove from heat and cool for 10 minutes.
Make the Cake Batter:
In a stand mixer, beat the browned butter and both sugars for 2 minutes. Add eggs and vanilla; mix 2 more minutes. Add Greek yogurt and milk, mixing to combine.
Add dry ingredients (flour, baking soda, baking powder, salt) and mix until just combined.
Bake:
Pour batter into a lined 8x8-inch pan. Bake at 350°F for 45–48 minutes, or until golden and a toothpick comes out clean. Let the cake cool completely.
Make the Salted Caramel:
While the cake bakes, melt butter and brown sugar in a small saucepan over medium heat for 5 minutes. Slowly add heavy cream, stirring until a thick caramel forms (about 2 minutes). Remove from heat and stir in vanilla and salt. Let cool.
Make the Frosting:
Beat butter and cream cheese on high for 2 minutes until fluffy. Add confectioners’ sugar, vanilla, and cinnamon; beat 2 minutes more.
Assemble:
Once the cake is cool, poke holes all over with a skewer or fork. Pour caramel sauce over the cake and let it soak in. Spread frosting on top.
Serve:
Cut into squares. Finish with extra caramel or flaky salt if you’re feeling fancy.
Notes
Tips & Notes:The cake is excellent served cold—just cover and refrigerate.For extra texture, sprinkle chopped toasted pecans on top.Caramel can be made ahead and refrigerated; just warm slightly before pouring.