Mash the bananas. In a medium mixing bowl, mash the bananas until mostly smooth.
Add the eggs. Crack in the eggs and whisk together until fully combined and slightly frothy.
Mix in cocoa powder. Stir in the cocoa powder until the batter is smooth and evenly mixed.
Prepare the mugs. Lightly grease two large (12–14 oz) microwave-safe mugs. Divide the batter evenly between them, filling no more than halfway.
Microwave one at a time. Cook each mug separately for 1 minute. Check the top, it should be set but still soft in the middle. If needed, microwave for another 10–15 seconds. (Don’t overcook.)
Make the topping. In a small bowl, combine the chocolate squares and milk. Microwave in 20-second intervals, stirring between each, until smooth and melted.
Finish and serve. Pour the warm chocolate topping over the mug cakes while they’re still hot. Serve immediately!