Prepare the Apricot Filling: Combine the chopped dried apricots, water, sugar, and lemon juice in a small saucepan. Bring to a simmer over medium heat. Cook, stirring occasionally, until the apricots have softened and the mixture has thickened into a spreadable jam consistency (about 8-10 minutes). Remove from heat and let cool completely.

Make the Cookie Dough: In a large bowl, cream together the softened butter and powdered sugar until light and fluffy. Beat in the vanilla extract and eggs one at a time until fully incorporated.

In a separate medium bowl, whisk together the flour, baking powder, and salt. Gradually add the dry ingredients to the wet ingredients, mixing on low speed until just combined. Do not overmix. The dough will be soft.

Assemble the Cookies: Preheat your oven to 350°F (175°C). Lightly flour a surface and roll the dough out to about 1/8-inch thickness. Use a 3-inch round cookie cutter to cut out shapes. Place a small teaspoon of the cooled apricot filling onto the center of half the dough circles. Top with the remaining circles, pressing the edges gently to seal the 'pillow'.

Bake the Cookies: Place the sealed pillows onto baking sheets lined with parchment paper. Prick the tops once lightly with a fork to allow steam to escape. Bake for 10-12 minutes, or until the edges are set but the centers remain pale and soft.

Cool and Finish: Let the cookies cool on the baking sheet for 5 minutes before transferring them to a wire rack to cool completely. Once cool, generously dust the Soft Sweet Apricot Pillow Cookies with remaining powdered sugar before serving.