Bring a pot of water to boil. Score the bottoms of the apricots with an "X", making the cuts halfway up the sides and through the skin.
Add the apricots to the boiling water for 30 to 60 seconds, just until the skin pulls away from the flesh. Quickly transfer apricots to a bowl filled with ice cubes.
Drain and peel the apricots. Cut in half, pit them and slice into ½-inch pieces. Puree in a blender for 1 to 2 minutes, until smooth. Run the puree through a fine-mesh strainer into a bowl.
In a small saucepan, combine the apricot puree, sugar, lemon juice, and ginger. Simmer over medium heat, skimming foam as needed until the puree reaches 215°F to 220°F (101ºC to 104ºC). Remove from the heat and cool.
Place the apricot filling in a ziplock bag and cut a small corner off right before decorating.
Set the oven rack to the center position. Preheat to 350ºF (177ºC).
Line two large sheet pans with parchment paper.
Cream the butter and sugar in a stand mixer on medium speed until light and fluffy, about 1 minute.
Add salt, vanilla, and egg. Beat well on medium speed for 1 minute. Scrape down the sides as needed.
Gradually add the flour, beating until just blended. The dough should be firm but neither sticky nor stiff.
Transfer the dough to a piping bag fitted with a large star tip (824 tip size).
Pipe the dough into small rounds on a parchment paper-lined baking sheet. The cookies should be about 1 ½ to 2-inches in diameter. Leave 2-inches of space in between.
Dip your finger into a bowl of water, and make small indents into the center of each cookie for the jam filling.
Pipe a small amount of jam into each indent of the cookie. If using cold jam, spoon in ¼ teaspoon into the center.
Bake one tray at a time, until cookies are lightly browned around the edges and on the bottoms, the surface will stay pale in color, 12 to 14 minutes.
Immediately transfer to a wire cooling rack.
If apricots are not in season, use a store-bought fruit jam of your choice. You will need about an 11-ounce jar.
Cookies can be stored in an airtight container for about five days, or frozen for up to 1 month.