Preheat the oven to 350°F (175°C). Grease a 9x13-inch baking dish.
Make the filling: In a small bowl, stir together 1/3 cup brown sugar, 1/4 cup cane sugar, and 1 teaspoon cinnamon. Set aside.
Make the crumble topping: In a medium bowl, combine 2/3 cup all-purpose flour (spooned and leveled), 1/3 cup brown sugar, 1/4 cup cane sugar, and 1 teaspoon cinnamon. Add 5 tablespoons unsalted butter (at room temperature) and use a spatula to press and mix until there is no dry flour remaining and the mixture is crumbly. Do not overmix. Set aside.
Whisk the dry cake ingredients in a large bowl: 2 1/2 cups plus 2 tablespoons all-purpose flour (spooned and leveled), 1 1/2 teaspoons baking powder, 1 1/2 teaspoons baking soda, and 3/4 teaspoon sea salt. Set this dry mixture aside.
In the bowl of a stand mixer or a large bowl with an electric mixer, cream 3/4 cup unsalted butter (at room temperature) until smooth.
Add 1 1/2 cups cane sugar to the creamed butter and beat until combined and slightly fluffy.
Add the 3 large eggs one at a time, beating well after each addition. Stir in 1 1/2 teaspoons vanilla extract.
Add the dry mixture and the 1 1/2 cups whole milk Greek yogurt to the butter mixture in alternating additions: add one quarter of the dry mixture and mix just to combine, then add one third of the yogurt and mix. Repeat this sequence (dry, yogurt, dry, yogurt, dry, yogurt), ending with the remaining quarter of the dry mixture. Mix only until the ingredients are incorporated; do not overmix.
Spread half of the batter evenly in the bottom of the prepared 9x13-inch pan.
Evenly sprinkle the prepared filling (from step 2) over the batter layer.
Dollop the remaining batter in spoonfuls over the filling, then use a spatula or an offset knife to carefully spread it into an even layer without mixing it into the filling.
Evenly sprinkle the crumble topping over the top of the cake.
Bake for 45 to 50 minutes, or until a toothpick inserted into the center comes out clean.