Make the dough: In a large bowl, whisk the rye flour, all-purpose flour, and salt. Add the melted butter and ¾ cup plus 1 tablespoon water; stir with a fork until a firm dough forms. Turn the dough out onto a clean surface and knead until smooth, 2–4 minutes. Cover with a dry towel and set aside.
Make the filling: In a heavy-bottomed saucepot, combine the milk and rice; bring to a simmer over medium-high heat. Immediately lower heat to maintain a gentle simmer, and cook, stirring occasionally, until the rice is tender, about 20 minutes. Stir in the melted butter, salt, and pepper, and remove from the heat. Once the mixture is slightly cool, cover surface with plastic wrap.
Meanwhile, prepare the egg butter for topping: In a medium saucepan, cover the eggs with cold water and set over high heat. Bring to a boil, then immediately turn off the heat and set a timer for 9 minutes. Drain the eggs and immediately plunge into ice water to cool completely. Peel the cooled eggs, finely chop, and add to a medium bowl. Add the softened butter, season with salt to taste, and stir to combine.
Preheat the oven to 450°. Divide the dough into 16 equal pieces (about 1½ oz. each). Lightly dust a clean work surface with rye flour, then, using a rolling pin, roll each portion into a circle about 7 inches in diameter. Using a 6-inch round cookie cutter or cake pan, trim to a perfect circle. Spread 3 tablespoons of the rice filling in an oblong shape down the center of each dough circle. Bring the two long outer edges of the dough to the center (the top of the rice filling should remain exposed), and press and pinch in places to create a scalloped, open, oval shape.
Lightly dust a baking sheet with rye flour. Place the pies on the sheet and generously brush the top of each with some of the melted butter.
Transfer to the oven and bake, brushing the pies with butter again halfway through cooking, until the pastries are crisp and darkened at the edges, 25 minutes. Immediately after removing from the oven, brush the pies with any remaining melted butter. Serve warm with a healthy dollop of egg butter.