Preheat oven to 350°. Line 9 x 13 baking pan with parchment paper.
Beat together 2 cups flour, butter and 1/2 cup sugar using stand mixer with paddle attachment or mix in a food processor. Sprinkle the dough into prepared pan and press down into an even layer. Bake until golden, about 20 minutes.
In large bowl, whisk together remaining 1/3 cup flour and 1-1/2 cups granulated sugar. Whisk in eggs, lemon juice, vanilla extract and lemon zest until completely combined. Pour mixture over baked crust.
Transfer pan to over and bake 25 minutes. Filling should be firm around edges and slightly soft in center. Remove from oven and allow to cool for 1 hour before covering pan and chilling for at least 4 hours. Dust with powdered sugar and cut into bars with sharp knife. Clean knife after each cut.
Store leftover bars covered in the refrigerator for up to 1 week.