In a medium bowl, throughly stir together the flour, baking powder, baking soda, and salt; set aside.
In a large bowl, with an electric mixer on low speed, beat together the butter, oil, and powdered sugar until blended.
Increase the speed to high and beat until very well blended.
Add the egg, lime and lemon zest, lime juice, vanilla and lemon extract and beat until lightened and smooth. Beat or stir in the flour mixture until evenly incorporated.
Refrigerate the dough for 1 1/2 hours or until firm enough to handle.
Preheat the oven to 350*.
Grease several baking sheets or spray with nonstick spray.
Shape portions of the dough into generous 1" balls with lightly greased hands. Place on baking sheets, spacing about 1 1/2" apart.
Bake the cookies, one sheet at a time, in the upper third of the oven for 13-16 minutes or until lightly tinged with brown at the edges. Reverse the sheet from from to back halfway through baking to ensure even browning.
Transfer the sheet to wire and let stand until the cookies firm up slightly.
Using a spatula, transfer the cookies to wire racks.
Let stand until completely cooled.
In a medium bowl, stir together the powdered sugar, lime zest, lime juice, corn syrup, and food coloring. Add a few drops of hot water if necessary to blend the ingredients together.
Stir until completely smooth.Thin with enough water to produce a fluid but not runny glaze.
Set the wire racks over wax paper to catch drips.
Working gently, since the cookies are fragile, swirl a layer of glaze over each.using a table knife.
Return the cookies to the racks.Let stand until the glaze completely sets, at least 1 hour.
Store in a single layer or layered with wax paper in an airtight container for up to 1 week or freeze up to one month --.
I sometimes use a clean paint brush to brush the glaze over cookies.works better for me than a knife.