Preheat the oven to 350°F (175°C) with a rack in the center position. Coat a 18-by-13-inch (45-by-33-cm) rimmed baking sheet with nonstick spray.
In the bowl of a stand mixer fitted with the paddle attachment or in a large bowl with a handheld mixer, beat the butter and sugar on medium-high speed until light and fluffy, 3 to 5 minutes. Add the eggs one at a time, beating well and scraping down the sides of the bowl after each addition (don’t worry if the mixture starts to look curdled; it will come together as you add more ingredients).
In a large bowl, whisk together the flour, baking powder, and salt.
In a large spouted measuring cup, whisk together the milk, oil, vanilla, and coconut extract.
Alternate adding the dry and wet ingredients to the egg mixture, starting and ending with the dry ingredients and mixing on low speed after each addition, then mix just until a smooth batter comes together.
Pour the batter into the prepared baking sheet and spread it evenly to the corners with a large offset spatula. Bake until the sides of the cake pull away from the pan and a tester inserted into the center comes out clean, 30 to 35 minutes.