In a mixing bowl, beat the eggs until foamy. Add sugar, salt, baking powder, anisette; beat until smooth.
In a small bowl, beat together oil and room temperature butter until well combined. Stir into batter.
Gradually add flour, beating only for a short time after flour has been added. Use more or less flour to make a soft cookie batter dough. (Using less flour will make for a lighter cookie). Dough should be soft, but stiff enough to form a smooth shape.
Cover; refrigerate dough for several hours.
Using a cookie scoop or two teaspoons, shape the dough into small balls; press bottom of cookies lightly onto lightly greased cookie sheets, while keeping the rounded tops.
Bake in a preheated 350°F oven for about 15 minutes. The cookies should be crisp but do not brown - they should be very light in color.
Using a spatula, transfer cookies to a wire rack to cool.
To ice the cookies: In a clean mixing bowl, combine 1 cup confectioners' sugar with 4 teaspoons milk and 4 teaspoons Karo syrup. Stir until the icing is of a spreadable consistency, adding more milk if required. A few drops of anisette may be added for flavor; a drop or two of food coloring can be stirred in for a pastel-colored icing, if desired.
Dip cookie tops into icing, then optionally sprinkle with colored cookie decorations.