For the topping:
Heat oven to 350 degrees F. Lightly grease and flour 12 standard-size donut cavities. 2 donut pans.
In a large bowl, whisk flour, sugar, rosemary, baking powder and salt until well combined.
In a separate medium bowl, whisk eggs, olive oil, milk, lemon zest and lemon juice until well combined.
Make a well in flour mixture, then pour in egg mixture. Use spatula to stir ingredients until just combined (batter may be lumpy -- this is OK).
Use a resealable plastic storage bag or piping bag to pipe batter into prepared donut cavities (you can also spoon the batter into the cavities, but it's a little messier).
Bake 11 to 14 minutes until a toothpick inserted in center of one donut comes out clean.
Let donuts cool in pan 5 minutes, then tap out onto sheet pan.
Place butter and sugar for topping into individual large shallow bowls.
Dip each warm donut into melted butter, then coat gently in sugar.
Serve immediately.
