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Place butter into a light-colored saucepan over medium-low heat. Cook, stirring constantly, until butter begins to smell nutty and browned bits begin to form in the bottom of the pan, 7 to 10 minutes. Remove from heat and pour browned butter into a heat-safe glass measuring cup. This should yield approximately 1 cup browned butter (it’s ok if it’s slightly more). Allow to cool for 15 minutes.
Preheat the oven to 350 degrees F (180 degrees C). Line an 8×8-inch square cake pan with enough parchment paper to have overhang on all sides.
For topping, stir flour, brown sugar, cinnamon, nutmeg, and salt together in a small bowl until thoroughly combined. Add 1/2 cup browned butter and maple extract. Mix together with a fork until crumbly and no dry bits of flour remain. Set aside.
For cake, combine remaining 1/2 cup browned butter, brown sugar,salt, cinnamon, and nutmeg in a large bowl. Whisk together until combined. Add eggs, maple syrup, Greek yogurt, vanilla, and maple extract and whisk until completely smooth and combined. Add flour, baking powder, and baking soda and mix until just incorporated and no large pockets of flour remain.
Pour batter into the prepared pan and spread into an even layer. Tap pan on the counter several times to pop any larger air bubbles in the batter. Sprinkle crumb mixture evenly over cake batter.
Bake in the preheated oven until a toothpick inserted into the center of the cake comes out with a few moist crumbs, 40 to 50 minutes. Cool cake in the pan for 10 minutes, then lift out using the parchment overhang to a wire rack to cool completely.