Preheat oven to 350 degrees F. Grease a 8x8-inch baking dish and line with parchment paper.
Make oat crumble. In a large bowl, combine flour, oats, brown sugar, granulated sugar, baking powder, and salt. Add melted butter and vanilla, stirring until well combined.
Add half of the crumble into the prepared baking dish and press down to create an even layer. Set aside remaining oat crumble. Bake crust for 15 minutes or until golden brown. Remove from the oven and let cool.
Reduce oven temperature to 325 degrees F.
Combine sugar and lemon zest together in a medium bowl, rubbing the sugar with the zest. Pour mixture into the bowl of a stand mixer with cream cheese and beat until creamy, about 2 minutes. Alternatively, used a hand mixer.
Add eggs one at a time. Add lemon juice, mixing until well combined. Pour filling in the baked crust and sprinkle remaining oat crumble on top.
Bake lemon cheesecake crumb bars for 40-45 minutes or until the edges have set, but the center is still slightly wobbly. Let cool on a wire rack for 1 hour. Transfer to a refridgerator and let cool at least 4 hours or overnight. Slice and serve.