Beat pressed dates and 1⁄2 lb. (2 sticks) of the butter together in a large bowl with an electric mixer on high speed until well combined and smooth, 2-3 minutes. Set filling aside.
Beat sugar and the remaining butter together in another large bowl with an electric mixer on high speed until light and fluffy, 1-2 minutes. Add eggs and vanilla and beat on high speed until well combined. Reduce speed to low and gradually add flour, beating until a smooth dough forms.
Turn dough out onto a lightly floured surface, divide into quarters, and roll each piece into a ball (chill dough in the refrigerator if it is too soft to work with). Roll dough balls out on the floured surface with your hands into four 1"-diameter cylinders. Dust your fingertips with flour and press center of each dough cylinder to make a 1⁄2"-deep indentation the length of each cylinder. Fill dough cylinders with one-quarter of the date filling. Work dough around filling (including ends of each cylinder) and pinch seems shut, completely closing filling. Sprinkle an even layer of sesame seeds onto a clean surface and roll dough cylinders in seeds, gently pressing seeds into dough for at least 30 minutes to chill.
Preheat oven to 375°. Cut each dough cylinder crosswise into 1"-wide pieces and transfer to a baking sheet, seed side down, about 1⁄2" apart. Bake until golden brown, 15-20 minutes. Set aside to cool completely.