For an egg-shaped cookie cutter, you can start with an oval cutter and bend one end into a slight point.
Whisk together flour, baking powder, and salt in a bowl. In another bowl, beat butter with sugars on medium speed until pale and fluffy, about 3 minutes. Beat in eggs and vanilla. Reduce speed to low; gradually beat in flour mixture. Divide dough in half; flatten into disks. Wrap each in plastic and refrigerate until firm, 1 hour or overnight.
Preheat oven to 325 degrees. Let dough stand at room temperature until pliable. Roll out on a lightly floured surface to just under 1/4 inch thick, dusting with flour as needed. Cut out shapes with cutters; transfer to parchment-lined baking sheets. Reroll scraps and repeat. Freeze until firm, about 15 minutes. Bake until golden, rotating sheets once, about 18 minutes. Let cool completely on sheets on wire racks.
For cookies with white and colored dots: Use a pastry bag fitted with a small round tip (such as #3) to make icing dots (about half as many dots as you want on finished cookies). Sprinkle with sanding sugar and let set, then shake off excess sugar. Pipe additional dots on cookies; let dry.
To coat cookie surface with icing: Place icing in a shallow dish and carefully dip cookie into icing, leaving a thin border. Transfer to a wire rack and sprinkle with sanding sugar, if desired. Let set and shake off excess sugar. If desired, pipe additional dots on cookies (or pipe a dot for a rabbit's eye). Let dry overnight. Cookies can be stored between layers of parchment up to 1 week.