Preheat the oven to 350 degrees F. Grease a 9- by 5-inch loaf pan.
Roughly chop 1/2 cup of the walnuts, and finely chop the remaining 2 tablespoons, keeping them separate.
Whisk together the flour, sugar, roughly chopped walnuts, baking powder, cinnamon, nutmeg and salt in a medium bowl.
Whisk together the oil, eggs, vanilla and zest. Fold in the bananas and strawberries. Fold the oil mixture into the flour mixture until just combined (it's OK if there are some lumps). Transfer the batter to the prepared pan, and sprinkle the top with the finely chopped walnuts.
Bake until the bread is golden brown and a toothpick inserted in the center comes out clean, 1 hour to 1 hour 10 minutes. Run a knife around the edges, and let cool completely in the pan on a rack. The bread can be stored at room temperature in an airtight container for up to 3 days.
When measuring flour, we spoon it into a dry measuring cup and level off the excess. (Scooping directly from the bag compacts the flour, resulting in dry baked goods.)
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