Preheat the oven to 350 degrees and butter a 9 x 5 loaf pan. Set aside.
Whisk the all-purpose flour, rye flour, baking soda, baking powder and salt together in a large bowl. Set aside.
Place the butter in a large skillet (preferably not nonstick or cast iron so that you can more easily see the color change) and heat over medium heat, swirling the pan frequently. Once the butter has melted, continue to cook over medium low, swirling the pan frequently, until the butter has a nutty aroma and is golden brown in color. Transfer brown butter to a medium-sized mixing bowl.
Add the cardamom and mashed bananas and whisk until homogenous in texture. Add the brown sugar and vanilla extract and whisk until thoroughly combined. Add the eggs, one at a time, whisking thoroughly after each addition.
Add the wet ingredients to the dry ingredients and fold together with a rubber spatula until the batter just comes together, taking care not to overmix.
Pour the batter into the prepared pan and sprinkle the top with turbinado sugar. Bake for 60-75 minutes or until loaf registers 200 degrees F on an instant read thermometer or a tester inserted into the middle of the bread comes out clean. Cool to room temperature before slicing and serving.
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Notes
To store: The banana bread will keep at room temperature for 3 days or tightly wrapped in the refrigerator for 5 days. If you want to retain the crisp texture on top of the loaf, store the bread at room temperature and place a piece of foil over the cut edge and around the sides leaving the top open.To freeze: Wait until the loaf has completely cooled to room temperature. Wrap in a layer of plastic wrap and then a layer of foil. Freeze for up to 3 months. You can also freeze individual slices - Slice the cooled bread and wrap each piece in plastic wrap.Adapted from Milk Street and Smitten Kitchen.