In a large bowl, use an electric mixer to beat the butter at medium speed until creamy; gradually add the brown sugar, beating until light and fluffy. Add the egg and vanilla, beating until blended.: The kitchen will fill with a sweet, buttery aroma as you beat the butter and brown sugar , and you should notice the mixture lighten in color and become airy, which means enough air has been incorporated for tender cookies. This aeration affects texture more than you might expect, creating tiny pockets that melt in your mouth. A common mistake is over mixing once the egg is added, which can cause a tougher final cookie, so stop when the ingredients are just combined. If the mixture looks greasy instead of creamy, the butter may have been too soft; chill the bowl briefly before continuing.
Add the flour and salt to the butter mixture, beating well. Divide the dough in half, and wrap in wax paper or plastic wrap. Chill 1 hour.: After adding the all purpose flour and salt , the dough will transform from glossy batter to a soft, pliable mass. You will notice the texture become more dough like as the flour absorbs moisture. Chilling firms the dough so it is easier to roll and helps the cookies keep their shape while baking, which preserves that delicate edge and tender center. A typical error is skipping the chill, which leads to spreading and misshapen cookies. If your dough is sticky after chilling, lightly flour the surface when rolling rather than adding more flour to the dough.
On a lightly floured surface, roll half of the dough to 1/8-inch thickness. Keep the remaining dough chilled until ready to use. Cut with a 3 1/2-inch round cookie cutter. Transfer the cookies to lightly greased baking sheets or baking sheets lined with parchment or silpat mats.: As you roll, the dough should feel cool under your palms and slightly tacky to the touch but not sticky, and you will see a consistent, thin sheet form. Rolling to a uniform 1/8 inch thickness is essential for even baking so the cookies crisp at the edges while staying tender in the middle. When cutting circles, press straight down and lift gently to avoid distorting the shape; ragged edges are a sign of too much handling. If the dough warms while rolling, pop the sheet into the refrigerator for 10 minutes to re chill before cutting more shapes.
Roll the remaining half of the dough to 1/8-inch thickness on a lightly floured surface. Cut with a 3 1/2-inch round cookie cutter. Transfer the cookies to prepared baking sheets. Cut out the center of each cookie with a 1- or 1 1/2-inch star-shaped cookie cutter (or any other shape).: Working with the second half, you will repeat the same rhythmic motion, and when you cut the smaller star shapes from the centers you will see negative space that will later cradle the seedless strawberry jam . The small cutouts should be tidy because jagged cutouts can cause uneven baking or jam leakage. One mistake is trying to reuse scraps too many times, which makes the dough tough. Gather scraps gently, chill briefly, and re roll once for best results.
Bake at 350°F for 5 to 6 minutes or until the cookies are lightly browned. Cool slightly on the baking sheets; remove to wire racks to cool completely.: While baking, the kitchen will get a warm, sweet scent. Watch for just the faintest golden color at the edges, a sign the cookies are done; over baking will dry them out. The cookies continue to set as they cool, so letting them rest on the sheet for a short time helps them firm without cracking. A common pitfall is leaving them on the hot sheet too long, which can cause the bottoms to brown too much. Transfer to wire racks so air circulates and they cool evenly without becoming soggy.
Spread the solid cookies evenly with strawberry jam. Dust the cookies with the cutout centers lightly with powdered sugar. Top the solid cookies with the sugar-dusted cookies, pressing lightly together to fill cutouts with jam.: When assembling, the glossy seedless strawberry jam should be thick enough to stay in place, offering a bright contrast to the pale cookie. Dusting the cutout tops with powdered sugar adds a soft sweetness and an elegant look. Press gently so the jam fills the window without oozing out the sides. A common mistake is using too much jam, which causes the cookies to slide or become messy; use a spoon or small offset spatula and a measured amount for consistency.