Preheat the oven to 350°F (177°C). Grease an 8½×4½-inch or 9×5-inch loaf pan with nonstick spray or butter. Line it with a parchment sling for easy removal if you like.
Toss the chopped strawberries with 1 tablespoon of flour in a small bowl until lightly coated. Set aside. This step keeps the berries from sinking to the bottom.
In a large bowl, whisk together the 2 cups flour, baking powder, baking soda, and salt until evenly combined.
In a medium bowl, whisk the egg, granulated sugar, and brown sugar until smooth. Add the buttermilk, oil, and vanilla. Whisk until fully blended.
Pour the wet mixture into the dry ingredients. Stir gently with a spatula just until no dry flour remains. The batter will be thick. Gently fold in the floured strawberries. Avoid overmixing—stop as soon as the berries are distributed.
Scrape the batter into the prepared pan and smooth the top. Bake for 55–65 minutes. About halfway through, loosely tent the pan with foil if the top is browning too quickly. The bread is done when a toothpick inserted in the center comes out clean or with a few moist crumbs, and the top feels springy.
Let the loaf cool in the pan for 15–20 minutes. Then transfer it to a wire rack to cool completely before glazing or slicing. Warm bread is delicious but can crumble more easily.
Whisk the powdered sugar, 1 tablespoon milk, and vanilla until smooth. Add more milk a teaspoon at a time if needed for a drizzle consistency. Pour over the cooled loaf and let it set for 10–15 minutes.