Preheat the oven to 350 degrees Fahrenheit. Line a large baking sheet with parchment paper.
Dry ingredients. In a large bowl, combine flour, baking powder, and salt. Set aside.
Wet ingredients. In a separate bowl or stand mixer with paddle attachment, whisk the eggs, sugar, vanilla until the sugar is dissolved in the mixture. About 2 minutes.
Stir in the olive oil and mix to combine.
Combine. In batches, add the flour mixture to wet ingredients, and beat on low with an electrical mixer until all the ingredients are combined. Dough will be relatively soft.
Sugar cookie dough. Transfer the dough onto lightly floured silicon baking mat or even surface. Knead the dough into a ball and with a rolling pin, roll the dough out flat to about ¼ inch thickness. Note: It doesn’t matter if the dough is irregular in shape or size. There is no need to chill the dough for these sugar cookies because there is no butter added.
Decoration. Using a cookie cutter, cut out the dough into desired shapes. Transfer each cookie onto prepared baking sheet.
Re-roll the remaining dough and continue cutting until all is used. Repeat with the rest of the dough by rolling it again into ball and then flat.
Bake the cookies for 11-12 minutes. The cookies are ready when the edges start to become golden. Allow cookies to cool on the baking sheet for 5 minutes then transfer to a wire rack to cool completely before decorating.
Simple icing. While the cookies are cooling, prepare the icing. Combine sifted icing sugar and milk. Feel free to tint the mixture with gel food coloring. Transfer the glaze into a pastry piping bag and decorate the cookies. Add sprinkles if using and let everything set.
When cutting the cookie shapes, try to cut them as close to each other as possible. You get the maximum cookie amount with less effort and re-rolling.
Enjoy cookies right away or wait until the icing sets to serve them. Plain or decorated cookies will stay fresh and soft about 4-5 days in an airtight container at room temperature. You can also refrigerate them for upto 10 days.