Misshapen Brownie

If two people want equal shares of the remaining brownie, then the line that goes through the middle of the original brownie rectangle and the middle of the hole rectangle will cut the remaining brownie area in half.

Proof: Any line through the middle of a rectangle cuts that rectangle in half, due to rotational symmetry. Therefore, the line through both the middle of the brownie rectangle and the middle of the hole rectangle will result in each person getting half of the original brownie area and half of the hole area. Therefore, each person will get equal amounts of the misshapen brownie area.

It can also be shown by continuity. Informally: integrate area from left (x=0) to right (x=a) . This is clearly continuous, and has values from 0 to whole area, so by the intermediate value theorem the result follows

Following the expression in the question, "vertical cut", I think it implies to divide the remaining part of brownie vertically. It should be possible, and thanks to Ville Karlssons (https://brilliant.org/profile/ville-ndy6hy) for the idea of intermediate value theorem, it was on my mind too, when I arrived at my solution. Measure the length and width of cut part and whole brownie, work out the areas. Subtract smaller area from the larger area. Half the difference. Divide the result by the width of tray (actual brownie) and get the length, which you need to measure from the left edge, to cut off. In result left part (full rectangle) will have the same area as right part with cut hole in it. Square of ratio of lengths of final two parts should be the same as ratio of areas of these parts.