Sift flour with baking powder. In a large bowl, using an electric mixer on medium-high speed, beat butter, sugar and salt until light, about 3 minutes. Beat in eggs and vanilla. Reduce mixer speed to low; beat in flour mixture just until combined, scraping down sides of bowl as needed.
Turn dough out onto a floured work surface; cut in half. Pat each half into a 1-inch-thick disk. (Alternatively, form disks into 2 or 3 11/2- to 2-inch-thick logs.) Wrap in plastic; chill for 1 hour or up to 2 days.
Place racks in upper and lower thirds of oven and preheat to 375°F. Line 2 large baking sheets with parchment. Roll out 1 disk on a lightly floured surface to 1/4-inch thickness. Use cookie cutters to form desired shapes. (If you've formed logs, cut into 1/4-inch-thick slices.)
Using a small metal spatula, carefully transfer cookies to lined baking sheets, spacing about 2 inches apart. Sprinkle tops with sugar, if desired. Chill for 10 minutes. Bake until lightly browned around edges and firm in center, 12 to 15 minutes, rotating sheets halfway through. Let cookies cool on sheets on wire racks for 5 minutes, then transfer cookies to wire cooling racks to cool completely.
Repeat with remaining dough, rerolling scraps to use up as much as possible. Decorate cooled cookies with icing, decorating sugar and candies, if desired.
Pistachio-Cranberry Cookies: Pulse 1/2 cup unsalted pistachios and 1 cup dried cranberries with 1 Tbsp. of the flour in a food processor to finely chop. Add to dough with rest of flour. Prepare cookies as recipe directs.
Peppermint Cookies: Substitute 1 1/2 tsp. mint extract for vanilla. Grind 20 small peppermint candies in a food processor (you should have about 1/2 cup). Add to dough with flour mixture. Prepare cookies as recipe directs, omitting optional sugar topping. Cool completely. Dip each cookie halfway into 8 oz. melted dark chocolate. Sprinkle with additional crushed mints, if desired. Chill until chocolate has hardened. Bring cookies to room temperature before serving.