For the dough: Dissolve yeast in 1 cup warm water in a small bowl and set aside until foamy, 5-10 minutes. Meanwhile, combine flour and salt in a large bowl. Add yeast mixture, 8 tbsp. of the oil, and grappa, and stir with a sturdy spoon until it holds together to form a dough. Shape into a rough ball, then transfer to a lightly floured surface. Knead dough, dusting work surface with a little flour as needed, until smooth, about 10 minutes, then shape into a ball. Grease a large bowl with remaining 1 tbsp. oil, place dough in bowl, cover tightly with plastic, and set aside in a warm place for 1-2 hours.
For the filling: Put anthotyro, feta, egg, and mint into a medium bowl and mix well. Set aside.
Preheat oven to 375°. Line several cookie sheets with parchment paper and set aside. Divide dough into 10 balls and flatten each into a rectangle on a lightly floured work surface. Working with one rectangle at a time, feed the wide side of the dough through the smooth cylinders of a hand-cranked pasta machine set on the widest setting. Decrease the setting by one notch and feed the narrow end of the dough through the cylinders again. Lightly flour dough as needed. Repeat, decreasing setting by one notch each time until you have used the narrowest setting. Lay sheets of dough out, not touching, on the lightly floured work surface, then cut into about 35 4''-5'' squares. Put 1 heaping tbsp. of filling in the center of each square. Working with one square at a time, fold each corner of the dough over the filling into the center of the square so that it resembles a little packet, overlapping tips of the corners in the center by a little bit. Use your fingertips to gently flatten cheese packets slightly.
Arrange cheese packets on prepared cookie sheets so that they don't touch each other, brush each with egg, then sprinkle with some of the sesame seeds. Bake until lightly golden, about 15 minutes.