Preheat oven to 350 degrees. Grease 2 6-slot doughnut pans with butter.
In a bowl, beat together the melted butter and sugar until combined. Whip in the eggs, one at a time, then the yogurt, lemon juice, lemon zest, and vanilla. Add the flour, baking powder, baking soda, and salt. Stir until just combined. The batter should be very thick.
Transfer the batter into a piping bag, or a ziptop bag with a corner snipped off. Pipe into the prepared doughnut pans, filling 2/3 of the way full. Bake for 10-12 minutes until risen and an inserted toothpick emerges with just a few moist crumbs. Cool 5 minutes in the pans, then run a knife around the edges of each doughnut to loosen and transfer to a wire rack. Cool completely.
Slice each doughnut in half. Spread one side with 1 tablespoon of the lemon curd. Sandwich the two halves together.
In a bowl, whisk together the powdered sugar, lemon juice, and vanilla. If needed, thin the glaze with water or milk, 1 teaspoon at a time. The glaze should be thick and not runny.
Set a piece of parchment beneath the wire rack. Dip each doughnut into the glaze and place on the rack to let any excess slide off. Let the glaze harden for about 5 minutes, then enjoy!