Preheat the oven and prepare the pan: Preheat your oven to 350°F (175°C). Grease a 9-inch round cake pan (or a square one) and line the bottom with parchment paper for easy removal.
Make the streusel topping: In a small bowl, combine the brown sugar, flour, cinnamon, and salt. Add the cold butter pieces and use a pastry cutter or your fingers to mix until the mixture forms a crumbly texture. Set aside.
Make the coffee cake batter: In a medium bowl, whisk together the flour, baking powder, baking soda, salt, and cinnamon. In a separate large bowl, cream the softened butter, granulated sugar, and brown sugar until light and fluffy (about 2-3 minutes). Beat in the eggs one at a time, then add the Greek yogurt and vanilla extract and mix until smooth.
Combine wet and dry ingredients: Gradually add the dry ingredients to the wet ingredients, mixing just until incorporated. Be careful not to overmix.
Assemble the cake: Pour half of the batter into the prepared cake pan. Sprinkle half of the streusel topping over the batter. Add the remaining batter and top with the rest of the streusel mixture. Gently press down on the streusel to ensure it sticks into the batter.
Bake the cake: Bake in the preheated oven for 35-40 minutes, or until a toothpick inserted into the center of the cake comes out clean.
Make the glaze: While the cake is baking, whisk together the powdered sugar, milk or cream, and vanilla extract to make the glaze. Adjust the consistency by adding more milk or sugar as needed.
Cool and glaze the cake: Allow the cake to cool in the pan for about 10 minutes before transferring it to a wire rack. Once completely cooled, drizzle the glaze over the top of the cake.
Serve: Slice and enjoy your Greek yogurt coffee cake with a cup of coffee or tea!