Prepare the grill for indirect cooking over medium heat (about 400ºF).
In a large bowl combine the graham cracker crumbs, 4 tablespoons of the melted butter, and the egg yolk; mix well. Firmly and evenly press the mixture into the bottom of an 8-by-8-inch metal baking pan. Grill the crust over indirect medium heat, with the lid closed, until firm, 6 to 8 minutes. Remove the crust from the grill and set aside to cool for about 10 minutes. This will allow the crust to set.
In a small bowl combine the remaining 1 tablespoon melted butter and the brown sugar. Cut each banana in half lengthwise and leave the skins attached (they will help the bananas hold their shape on the grill). Liberally brush the cut side of the bananas with the butter mixture.
Grill the bananas, cut side down, over direct heat, with the lid open, until they start to soften, 2 to 4 minutes, without turning. Let cool briefly and then score the bananas into ½-inch slices, cutting through just to the peel.
Fill the baking pan with half of the marshmallows, making sure they cover the bottom of the crust evenly. Next, scoop out and distribute the slices of grilled banana, followed by the remaining marshmallows.
Place the pan over indirect medium heat (keep the temperature of the grill as close to 400ºF as possible), close the lid, and cook until the marshmallows have puffed up and started to brown, 5 to 7 minutes. At this point carefully sprinkle the chocolate chips over the top and continue cooking until the chips appear glossy and melted, about 2 minutes more. Remove the pan from the grill and let cool for about 5 minutes. Spoon the banana s'mores into small serving bowls and serve right away.