With an electric mixer, beat the butter, cream cheese, and granulated sugar until fluffy. Mix in the vanilla. Gradually add the flour, mixing until just incorporated.
Turn the dough onto a floured work surface and gently knead it 2 to 3 times, just to bring it together. Form the dough into two 1-inch-thick squares. Wrap in plastic wrap and refrigerate for 1 hour.
On a lightly floured piece of parchment, roll one of the dough squares into a 9-by-12-inch rectangle. Spread half the jam over the dough. Cut the dough crosswise into thirds, making three 9-by-4-inch rectangles. Starting from the long side of each rectangle, roll into logs. Wrap in wax paper and refrigerate until firm, 30 minutes. Repeat with the remaining dough.
Heat oven to 350° F. Slice the logs into 1-inch pieces and space 1 1/2 inches apart on parchment-lined baking sheets. Brush with the egg and sprinkle with the turbinado sugar. Bake until golden, 20 to 25 minutes.
To Wrap: A 32-ounce glass canning jar will neatly hold 2 dozen pinwheels. You can play up the cookies' two-tone theme with a striped ribbon tied around the jar's mouth and a colorful adhesive bow.
To Freeze: Instead of refrigerating the logs, freeze them for up to 2 months. To bake, follow the recipe instructions, cutting and baking the dough from frozen, and use the upper end of the time range.