Preheat the oven to 350°F. Line 2 large baking sheets with parchment paper.
In the bowl of a stand mixer or using a hand mixer, beat the butter, sugar, lemon zest, vanilla, and salt until very light and fluffy, about 3 to 4 minutes.
Beat in the egg until well combined. Stir in the flour until thoroughly mixed in.
Scoop about 1 heaping teaspoon batter for each cookie or use a teaspoon cookie scoop. Roll each scoop into a ball in your hands and place on cookie sheets about 2 inches apart. ( If you want larger cookies use a Tablespoon cookie scoop or a heaping Tablespoon).
Using your finger tip (or thumb for larger cookies) press an indentation into the center of each cookie ball. I also find that using the back of a more rounded 1/4 teaspoon measuring spoon works well on the cookies (a 1/2 or 1 teaspoon spoon would work for larger cookies).
Bake cookies in preheated oven for 12 to 14 minutes. They will still be soft and not browned on edges.
Allow to cool on sheet for a few minutes and remove to cooling rack to cool completely.
Once cooled, heat jam in microwave until thin and liquid and just simmering. This takes me about 45 seconds to 1 minute. Using a spoon, fill the indentations with the jam. Allow to rest until jam thickens and sets up.