Preheat and prep: Preheat the oven to 350°F (175°C). Grease a donut pan and set aside.
Combine the dry ingredients: In a medium bowl, whisk together the flour, baking powder, baking soda, and salt. Set aside.
Make the lime sugar: In a large mixing bowl, combine the granulated sugar with the lime zest. Rub the zest into the sugar using your fingertips until fragrant and well combined.
Add the wet ingredients: To the lime sugar, mix in the melted butter, vanilla extract, eggs, milk, sourdough discard, and fresh lime juice. Whisk until smooth and fully combined.
Mix the batter: Gradually add the dry ingredients to the wet ingredients. Stir until just combined—do not overmix.
Fill the donut pan: Spoon or pipe the batter into the prepared donut pan, filling each cavity about ¾ full.
Bake: Bake for 13 minutes, or until the donuts are lightly golden and spring back when gently pressed.
Cool: Let the donuts cool in the pan for 5 minutes, then transfer them to a wire rack to cool completely before glazing.
Make the glaze: In a bowl, whisk together the powdered sugar, lime juice, and milk until smooth. Adjust the consistency if needed by adding a little more milk or sugar.
Glaze and garnish: Dip the tops of each cooled donut into the lime glaze. Sprinkle with crushed graham crackers and a bit of extra lime zest for a key lime pie-inspired finish.