Preheat the oven to 350°F (175°C). Line an 8 x 8-inch baking dish with parchment paper, allowing excess parchment to extend beyond two opposite sides so the baked cake can be lifted from the pan easily. If parchment is unavailable, thoroughly coat the baking dish with nonstick cooking spray.
Place the softened Greek yogurt cream cheese, granulated sugar, egg, and freshly squeezed lemon juice in a medium mixing bowl. Stir until the ingredients are completely blended and the mixture has a smooth, creamy consistency. Set aside while preparing the cake batter.
In a separate medium bowl, whisk together the all-purpose flour, baking powder, baking soda, and salt until evenly distributed. Reserve this mixture for incorporation into the wet ingredients.
Add the vegetable oil and granulated sugar to the bowl of an electric stand mixer and mix until combined. Add the egg and continue mixing until fully incorporated.
Blend in the Greek yogurt, finely grated lemon zest, and lemon juice. Mix at medium speed until the wet mixture appears smooth and creamy.
Gradually add the prepared dry ingredients, mixing at low speed only until no visible streaks of flour remain. Avoid excessive mixing, as overworking the batter can result in a denser cake.
Transfer the lemon cake batter to the prepared baking dish. Spread it into an even layer, smoothing the batter toward the edges of the pan.
Carefully distribute the cream cheese mixture over the cake batter, spreading it almost to the edges to create a distinct creamy layer.
Combine the flour and granulated sugar in a medium bowl. Add the cold cubed unsalted butter.
Using a pastry cutter or two forks, work the butter into the dry ingredients until the mixture develops coarse, pea-sized crumbs. Scatter the crumble evenly over the cream cheese layer.
Bake at 350°F (175°C) for 40–45 minutes, until the surface is golden and a toothpick inserted near the center comes out clean.
Allow the coffee cake to cool in the baking pan. Once cooled, lightly dust the surface with powdered sugar, if desired.
Using the parchment overhang, lift the cake from the pan and place it on a cutting board. Cut into 16 evenly sized squares before serving.