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For batter: Preheat the oven to 350°F. Butter a 9x5-inch loaf pan and line it with parchment paper, if using. Combine the sugar, melted butter, oil, lemon zest, lemon extract, vanilla, and milk. Whisk well, to blend. In a separate bowl, whisk the eggs and buttermilk until evenly blended. Set aside. In another bowl, sift the flour, baking powder, and salt. Using a wooden spoon, stir a third of the dry mixture into the sugar mixture. Add half of the buttermilk mixture, stir until smooth, then add another third of the dry ingredients, the remaining buttermilk mixture, and the rest of the dry ingredients, stirring until smooth after each addition. Stir in the coconut. Scrape the batter into the prepared pan and smooth the top with a spoon. Bake on the center oven rack for about 50 minutes, until a tester inserted deep into the center of the bread comes out clean. The top will be deep golden brown. Cool the loaf in the pan on a rack for 10 minutes. Remove the loaf from the pan and cool to room temperature.

For glaze: Combine the confectioners’ sugar, lemon juice, melted butter, and lemon extract in a small bowl. Whisk well, to blend. The glaze should have the consistency of heavy cream. Adjust, as needed, with a little more sugar (to thicken) or drops of lemon juice or water (to thin). Do not use milk; it could curdle the glaze. When the bread has cooled to room temperature, spoon the glaze evenly over the top of the bread, then immediately sprinkle with the coconut. Garnish with lemon zest, if desired.