The diamonds can be made up to three days ahead and stored at room temperature.
Beat together butter, cream cheese, salt, and sugar with an electric mixer on medium-high speed until creamy. Reduce speed to low and beat in flour, 1 cup at a time, then vanilla.
Divide dough in half; form each half into a flat rectangle and wrap in plastic wrap. Refrigerate until firm but still pliable, about 45 minutes. Dough can be refrigerated up to 2 days. Dust 3 tablespoons sugar on a piece of parchment. Place 1 rectangle of dough on top and dust with remaining 3 tablespoons sugar. Roll out to just under 1/4-inch thick. Trim to a 12-by-16-inch rectangle and transfer on parchment to a baking sheet. Freeze until firm, about 30 minutes. Repeat with other rectangle of dough.
Preheat oven to 350 degrees. Slide 1 sheet of dough on parchment onto a work surface and evenly spread with preserves with a small offset spatula. Invert remaining sheet of dough on top; remove parchment and let stand until soft enough to cut but still firm. Trim edges, then cut horizontally into strips the width of a ruler. Repeat with same-size strips on the diagonal to create diamonds.
Freeze diamonds until firm, about 10 minutes. Transfer to baking sheets lined with clean parchment, spacing 1 inch apart. Freeze again until firm, about 10 minutes. Bake until golden, 20 to 23 minutes. Let cool on baking sheets 5 minutes, then dust with sugar. Let cool completely before removing from paper with a spatula. (Preserves may cause them to stick.) Diamonds can be made 3 days ahead and stored at room temperature.