While the crust is baking, place the whole eggs, egg yolk, granulated sugar, powdered sugar, and salt in a large bowl and whisk to combine. Whisk in the lemon juice, lime juice, mint, and flour until just combined (try not to overmix).
Remove the parbaked crust from the oven. Pour the filling onto the warm crust. Make sure to tap out any air bubbles on the counter before baking.
Return the pan to the oven and bake until light brown around the edges, set in the middle, and the top appears relatively dry, 15 to 20 minutes. Cool completely on a wire rack, at least 1 hour, before removing from pan.
Using a fine-mesh strainer, generously dust the bars with powdered sugar (see Recipe Notes).
Move the bars to a cutting board by grasping the parchment sling and lifting the slab out of the pan. Cut into 12 or 16 bars. Dust with more powdered sugar, if desired, before serving.
These bars are best eaten within 24 hours, but may be covered and refrigerated for up to 3 days.