Preheat the oven to 350°F (175°C). Grease a 9×5 inch loaf pan or line it with parchment paper.
In a large mixing bowl, mash the ripe bananas thoroughly with a fork until smooth.
Add the two large eggs to the mashed bananas and whisk together until fully combined.
Gradually add 1 1/2 cups of self-rising flour to the banana and egg mixture. Stir gently until just combined, being careful not to overmix.
Pour the batter into the prepared loaf pan and spread it evenly.
Bake in the preheated oven for 50 minutes, or until a toothpick inserted into the center comes out clean.
Remove the banana bread from the oven and let it cool in the pan for 10 minutes.
Transfer the bread to a wire rack to cool completely before slicing and serving.