In the bowl of a stand mixer fitted with the dough hook, combine the rye and all-purpose flours with the butter, 1 teaspoon salt, the eggs, and 3⁄4 cup lukewarm water. Knead on medium speed until the dough comes together and is smooth, about 6 minutes. Scrape the dough onto a lightly floured work surface and shape into a ball. Halve the dough and shape each half into a disk. Wrap each disk in plastic wrap and refrigerate for 30 minutes.
Meanwhile, grate the cooked potato on the large holes of a box grater and reserve 1 cup; discard the rest. Place the potato in a medium bowl, mix with the ricotta and scallions, and season with salt and pepper.
On a floured work surface, roll each dough disk into a 1⁄8-inch-thick circle. Drop 1-tablespoon-sized dollops of the ricotta-potato filling evenly spaced over 1 dough circle. Using a pastry brush, lightly brush the dough with water around each dollop of filling. Drape the second dough circle over the first and gently press the dough between the mounds of filling to adhere. Position a 3-inch-round fluted cutter over 1 mound of filling and stamp out the round. Repeat, stamping out all the rounds.
Pour enough oil into a 6-qt. saucepan to come 2 inches up the side, attach a deep-fry thermometer, and heat to 350°. Working in batches, add the rounds to the oil and fry, turning occasionally, until golden brown, about 2 minutes. Using a slotted spoon, lift the fritters from the oil and drain on paper towels. Season the fritters with salt and serve while hot.