Preheat the oven to 350ºF and grease a 9x5-inch loaf pan.
In a large bowl, whisk together the eggs, bananas, and butter, then stir in the rest of the ingredients, stirring until a smooth batter forms. Fold in the strawberries. Pour the batter into the prepared pan, smooth the top, then rest the batter until the oven is fully heated.
Bake the loaf for about 50-55 minutes, when a tester comes out clean or with a few moist crumbs.
Cool the bread in the pan for 20 minutes, then cool it on a rack. I like to cool the bread completely so that the crumb sets, but it's OK to cut into it after about 30 minutes (it's delicious warm!).
Notes
Can I use frozen bananas? Absolutely. A great way to save ripe bananas is to freeze them (skin on or off) in a freezer-safe bag. When you're ready to bake, take out as many bananas as you need, let them defrost in a bowl, and drain away any liquid before mashing them up for the recipe.
Rest the batter. Don't turn the oven on until you have assembled the batter. While the oven is preheating, the gluten-free flour will have time to hydrate (soak up the moisture in the batter), making for a softer, better crumb after the loaf is baked. About 15-20 minutes should do it.
Can I use another flour? You can use a single-origin flour like oat, sorghum, or brown rice flour instead of the oat flour by weight. Or try another single-origin gluten-free flour, like sorghum flour or brown rice flour, by weight.
Try other spices. Instead of cinnamon, use ground ginger or cardamom. Or, add fresh peeled and grated ginger!
Add vanilla. Stir in 2 teaspoons of pure vanilla extract or vanilla bean paste when putting together the batter.
Make it refined sugar-free. Substitute coconut sugar or maple syrup for the granulated sugar.
Dairy-free: Swap melted dairy-free butter or melted coconut oil for the unsalted butter.