Whisk together flour and salt in small bowl. Beat together butter and sugar in large bowl with electric mixer at medium-high until pale and fluffy, about 3 minutes in stand mixer or 6 minutes with handheld. Beat in egg and vanilla. Reduce speed to low and add flour mixture, mixing until just combined.
Form dough into 2 balls and flatten each into 6-inch disk. Chill disks, wrapped in plastic wrap, until firm, at least 1 hour.
Put oven rack in middle position and preheat oven to 350°. Roll out 1 disk of dough (keep remaining dough chilled) into 8 1/2-inch round (1/4-inch thick) on well-floured surface with well-floured rolling pin. (If dough becomes too soft to roll out, rewrap in plastic and chill until firm.) Cut out as many cookies as possible from dough with cutters and transfer to 2 ungreased large baking sheets, arranging cookies about 1 inch apart.
Bake cookies, 1 sheet at a time, until edges are golden, 10 to 12 minutes; transfer to racks to cool completely.
Meanwhile, gather scraps and chill until dough is firm enough to reroll, 10 to 15 minutes. Make more cookies with remaining dough and scraps (reroll scraps only once); bake on cooled baking sheets.
If using icing and coloring it, transfer 1/4 cup icing to small bowl for each color and tint with food coloring. Spoon each color icing into sealable bag, pressing out excess air; snip 1/8-inch opening in bottom corner. Twisting bag firmly just above icing, decoratively pipe icing onto cookies. Let icing dry completely (about 1 hour) before storing cookies. Makes about 2 to 3 dozen cookies.
Note: Dough can be chilled up to 3 days. Cookies keep, layered between sheets of wax paper or parchment, in airtight container at room temperature 1 week.
Beat together all ingredients in large bowl with electric mixer at medium speed until just combined, about 1 minute. Increase speed to high and continue to beat until icing holds stiff peaks, about 3 minutes in stand mixer or 10 minutes with handheld. If not using icing immediately, cover surface with dampened paper towel then cover bowl with plastic wrap.
Note: Icing can be made 1 day ahead and chilled, its surface covered with dampened paper towel and bowl covered with plastic wrap.
The understated, familiar flavor of these delicate cookies will take you straight back to your childhood.