Place the peeled and roughly chopped beets in a saucepan with boiling water. Boil for 15 minutes, then remove with a slotted spoon to cool 10 minutes. Reserve 1/4 cup of the beet water to add to a high speed blender with the beets. Blend until smooth.

Combine 3/4 cup of the pureed beets, egg, and salt in a large bowl. Mix well.

Add the flour and mix until a shaggy dough. Turn out the dough on a clean surface to knead until a smooth dough that's not sticky. Dust with flour as needed. Cover and let rest at room temperature for 30-60 minutes.

In the meantime, add the peeled and roughly chopped potatoes to a fresh pot of boiling water. Add 1 tablespoon salt and stir to combine. Reduce heat to a simmer to cook 10-15 minutes, until fork tender.

Remove the potatoes with a slotted spoon to a food processor, or if mashing by hand in a large bowl. Let the potatoes steam while you prep the onions.

Add the finely chopped onions and 2 tablespoons of butter to a large skillet over medium high heat. Sauté 5-6 minutes, then add 1/4 cup water. Continue stirring until golden brown and fragrant. Remove from the skillet.

Mash the potatoes after they've slightly cooled, then stir in half of the caramelized onions.

Divide the dough into 2 parts. Roll out each part to a 1" thick log. Slice the log into as many 1" pieces. Roll out on a floured surface each piece to 1/8" thick circle.

Add a generous amount of the potato filling, then fold in a half-moon shape. Pinch the edges together to seal in the filling. Repeat with remaining dough and filling.

Bring a large pot of salted water to a boil (you can use the potato water). Boil in small batches for 2-3 minutes, until the dumplings float to the top. Stir to prevent them from sticking to the bottom. Remove with a slotted spoon.

Add the remaining 2-3 tablespoons of butter to the skillet over medium heat. Stir until the butter forms brown curdles at the bottom and smells itty, about 6-8 minutes.

Add the dumplings to cook 2-3 minutes on each side to crispen up.

Enjoy warm with sour cream, the remaining caramelized onions, fresh dill, and lemon wedges!