Preheat your oven to 400 degrees F. Grease two donut pans with butter and flour and set aside. (If you only have one pan you can work in batches, just be sure to grease the pan before baking the second batch.)
In a medium mixing bowl, whisk together the flour, sugar, baking soda, and salt. Set aside.
Ina large mixing bowl, whisk together the melted butter, brown sugar, egg, and vanilla yogurt. Add the dry ingredients to the bowl. Stir until just combined with a wooden spoon. (The batter will be very stiff.) Fold in the lemon zest.
Transfer the donut batter to a disposable piping bag and pipe into the prepared donut pans. (see note)
Bake for 8- 10 minutes until donuts are golden brown. Allow donuts to cool for 5 minutes in the pan before transferring to a wire rack to cool completely.
Icing
In a medium mixing bowl whisk together the powdered sugar and lemon zest. Then whisk in the melted butter and lemon juice until the icing is smooth and creamy. (If you want a thicker icing, add more powdered sugar.) If you're using yellow food coloring, start with a drop and whisk. Add more until your desired color is reached.
Dip the tops of the cooled donuts in the icing and then return them to the wire rack placed over a baking sheet. This will catch any icing that drips down the sides. If you're using sprinkles add them on top of the donuts immediately. Then let the donuts rest until the icing is set.
Notes
The donut batter in this recipe is very thick. If you do not have a disposable piping bag, use a ziplock bag with the corner cut off to pipe the batter as it would be very difficult to spoon the batter into the donut pan.
These donuts are freshest the day they are made. The leftovers can be refrigerated in an airtight container for 1-2 days. They can be brought to room temperature before eating.