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Combine the flour and granulated sugar in a large mixing bowl then sprinkle in the diced butter. With a pastry blender, moving in an up and down and twisting motion cut the mixture multiple times until the mixture is uniform and resembles coarse crumbs.
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In a small bowl whisk the egg, vanilla and milk and add it to flour/butter mixture. With a spatula, stir to combine, until a thick ragged dough starts to form. Add 1 tablespoon more milk if necessary.
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With floured hands, knead the dough in the bowl until it becomes one uniform clump of dough. Divide the dough into two equal portions, form 2 balls and flatten into 4” – 5” disks. Wrap each disk tightly in plastic wrap and refrigerate for at least 2 hours or overnight.
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Bring refrigerated dough to room temperature for 10 – 15 minutes. Meanwhile, preheat oven to 350 degrees F. Prepare 3 large baking sheets by lining with parchment paper.
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Lightly flour a clean work surface. Roll out one disk at a time to about ⅛” – ⅙” thickness. While rolling sprinkle the top of dough lightly with more flour as needed.
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Cut desired shapes with cookie cutters. For half of the shapes, cut out a small circle hole in the centre (an apple corer is just the right size for the hole). Re-roll the scraps, repeating the process to make more cookies until all the dough is used. If desired, with some of the re-rolled scraps, cut out mini shapes to use as an accent when displaying the cookies (these mini shapes won’t be filled).
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Transfer the cookies to the prepared baking sheets. If making mini cookies, place them on a separate baking sheet as they will bake a little quicker.
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Bake for 8 - 10 minutes (less for mini cookies) to a light golden colour; do not over-brown. Remove from oven and let cookies set on baking sheet for 2 minutes before removing to a rack to cool completely.
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To assemble the cookies, spread the “underside” (bottom baked side) of the solid cookies with about ½ teaspoon of jam, spreading almost to the edges. Cover with the cookies with a hole (placing the hole cookies baked side down), taking care to match the cut-out pattern. Press down very lightly.
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Transfer filled cookies in one layer to a parchment lined surface. With a sifter or a small sieve, lightly dust the tops with icing sugar. Store cookies in an air-tight container, with sheets of parchment paper between layers, for 12 – 24 hours before serving.