Preheat oven to 375°. Line cookie sheet with parchment paper.
In a saucepan, combine strawberries and ¼ cup sugar mashing strawberries to release juices. Let sit for about 5 minutes. Cook on medium heat, stirring occasionally and crushing strawberries until thick and jam like. Remove from heat to cool.
In a medium bowl, beat granulated sugar and butter with a mixer at medium speed until well-blended, 3-5 minutes. Add 1 ½ teaspoons vanilla and egg white, beat well for 1 minute.
In another bowl, combine flour, cocoa powder, cornstarch, baking powder, and salt, stirring well with a whisk. Add flour mixture to sugar mixture, stirring until well-blended.
Turn dough out onto a lightly floured surface. Divide dough in half. Roll each portion into a 12-inch log. Place both logs 3 inches apart on a baking sheet . With the end of a wooden spoon (or finger) form a trough down the length of each log.
Spoon strawberries into the troughs of each log.
Bake at 375° for 20 minutes. Remove logs to a cutting board.
Combine powdered sugar, milk and extract to form a thin glaze. Drizzle sugar mixture over warm logs. Immediately cut each log diagonally into 12 slices. Do not separate slices. Let cool for 10 minutes, separate slices and cool completely.
(Do not separate slices.) Cool 10 minutes; separate slices. Transfer slices to wire racks. Cool completely.