In a mixer, cream butter with the paddle attachment on medium speed until light and fluffy, about 3 minutes, scraping bowl and beater as needed. Gradually beat in 2 cups sugar, then beat until fluffy again, about 10 minutes. Reduce speed to low and gradually beat in flour, scraping bowl and beater a couple of times (dough will be uniformly crumbly). Add whole eggs and beat until smooth.
Turn dough out onto a work surface and roll into 2 logs, each about 2 in. wide. Roll in plastic wrap and chill at least 1 hour to firm up. Meanwhile, preheat oven to 375°.
Slice dough into 1/8-in.-thick slices, then arrange slices on parchment-lined baking sheets. Mix remaining 1/4 cup sugar with the cinnamon. Brush dough slices with egg white, then sprinkle with the cinnamon sugar. Push a few -almond slivers into center of each slice.
Bake until pale golden on top, about 10 minutes (if you like them completely crisp, cook until browned underneath, a couple minutes more). Remove and let cool on sheets until firm, about 10 minutes. Then transfer to wire racks to cool completely.