Step 1

Line a fine-mesh sieve with cheesecloth. Melt butter in a medium heavy saucepan over medium heat, stirring often, until simmering. Cook, stirring constantly, 10 minutes. Reduce heat to low; continue cooking, stirring constantly, until golden brown and you can see bits of browned caramelized milk solids on your spatula, about 10 minutes longer. Strain ghee through prepared sieve into a large jar. Set aside brown-butter ghee and caramelized milk solids in sieve separately.

Step 2

Whisk flour, brown sugar, cardamom, cinnamon, ginger, nutmeg, cloves (if using), star anise (if using), and salt in a medium bowl. Add 1 Tbsp. reserved caramelized milk solids and ½ cup reserved ghee. Using your hands, work into dry ingredients until mixture resembles wet sand.

Step 3

Do ahead: Brown-butter ghee can be made 3 months ahead. Store tightly covered at room temperature.

Step 4

Preheat oven to 325°. Line a 9x9" baking pan with parchment paper, leaving overhang on 2 sides. Whisk flour, baking powder, baking soda, and salt in a medium bowl. Whisk eggs, egg yolk, yogurt, milk, granulated sugar, brown sugar, vanilla extract, ¾ cup reserved ghee, and 1 tsp. reserved caramelized milk solids in a large bowl to combine. Sift in dry ingredients and fold just until no dry spots of flour remain (batter will be thick and slightly lumpy). Spoon half of batter into prepared pan; spread to edges. Sprinkle half of streusel evenly over, then dollop remaining batter on top and gently spread into an even layer. Sprinkle remaining streusel on top and gently pat to adhere to batter.

Step 5

Bake cake until a tester inserted in the center comes out clean, 40–45 minutes. Transfer pan to a wire rack; let cake cool before slicing.

Step 6

Do ahead: Cake can be baked 4 days ahead. Store tightly wrapped at room temperature.