Preheat the oven to 350ºF (180ºC). Line 2 baking sheets with parchment paper and line a large bowl with plastic wrap.
In a medium bowl, mix 1 package of the brownie mix, ⅓ cup (80 ml) oil, ⅓ cup (80 ml) water, and 1 egg until just combined. Pour the mix onto a prepared baking sheet and spread with a spatula until even. Repeat with the remaining brownie mix, oil, water, and egg. Bake for 13-15 minutes, until the brownies are fudgy, yet firm. Remove from the oven and let cool.
Once cool, cut 1 pan of the brownies in half lengthwise, then into 8 equal rectangles. Cut 4 of those rectangles diagonally to make a total of 8 triangles and 4 rectangles. Configure the brownie pieces in the lined bowl so that they cover the sides and the bottom. Once the inside is covered, press down firmly on all the brownies, especially the seams, to create a shell. Make sure there are no openings in the outer layer.
Scoop the strawberry ice cream onto the brownie shell and smooth with a spatula. Cover with plastic wrap and press a medium bowl into the ice cream so that it rises up on the sides, until flush with the top of the bowl. Freeze for 3 hours, until the strawberry layer is solid.
Unwrap and add the vanilla ice cream. Cover with plastic wrap and press a small bowl into the ice cream so it comes up the sides and is flush with the strawberry layer. Put back in the freezer for 2 hours.
Meanwhile, cut a circle in the other pan of brownies the circumference of the large bowl. This will cover the ice cream layers, serving as a base when the bomb is flipped over.
Uncover the frozen vanilla layer and fill the well with the chocolate ice cream. Cover with the brownie circle (save brownie leftovers for later!). Press the circle into the bowl to secure it, then freeze for 1 hour.
Carefully remove the bomb from the freezer and flip it over onto a cutting board or serving platter. It may need to thaw a couple minutes before it dislodges from the bowl. (Running a warm towel around the bowl will also do the trick.). Once you can lift up the bowl easily, uncover the bomb and dust with cocoa powder.