Combine the blue cheese and butter in a small food processor; process until creamy. In a small bowl, combine the flour, cornstarch, pepper, and kosher salt together; add to the blue cheese mixture. Pulse to combine. Add the chopped walnuts and process just until incorporated. Do not over-process. Remove the blue cheese mixture from the food processor bowl and shape into a ball; cover with plastic wrap. Refrigerate until firm, at least one hour.
Preheat the oven to 325°F.
Place the chilled dough on a piece of plastic wrap and cover with another piece of plastic wrap. Roll the dough out about ⅛-inch thick. Remove the plastic wrap and cut into 1-inch circles or rectangles using a fluted cookie cutter. Place on a parchment paper-lined baking sheet. Repeat until all dough is used. Bake until light brown, about 25 minutes. Let cool.
To assemble: Spread ¼ teaspoon of the cream cheese on each shortbread. Top with equal amounts of chutney, a walnut half, and a leaf of parsley.
Notes
To toast the walnuts, preheat oven to 350 degrees. Spread walnut halves on a small baking sheet and toast in the oven until scented and lightly browned. Watch them closely so they don't burn!
Do ahead tip: The shortbread can be prepared up to 3 days in advance and stored in an airtight container. Assemble up to 1 hour before serving.