In a bowl, combine the eggs, butter, sugar, lemon zest, lemon juice, and anise extract. With an electric mixer, beat the ingredients until well combined, about 3 minutes. Add the flour, baking powder, baking soda, and salt; beat on low to combine. Gather up the dough into a ball, wrap in plastic wrap, and refrigerate for 30 minutes.
Preheat the oven to 350 degrees F.
Line 2 baking sheets with parchment paper.
On a lightly floured surface begin to form the baskets.
To form the basket handle, start with a piece of the dough about the size of a walnut. Gently roll the dough into a thin pencil shape about 10 inches long.
Repeat with another piece of dough.
Place the 2 lengths side by side and twist one over the other like a spiral.
Place the handle onto a baking sheet in a horseshoe shape.
Take about 1/4 cup of the dough and form a thick oval or egg shape. Place one of the hard boiled eggs in the center of the oval. Press it down gently to embed it in the dough but don't push it through the dough.
Place the dough with the egg on top of the ends of the basket handle.
Press lightly to adhere the shape to the basket handle.
Continue making the baskets in this manner.
About 4 basket cookies should fit on each baking sheet.
Bake for 20 minutes until very slightly golden brown.
Transfer the cookies to a wire rack to cool completely.
To make the icing:
In a small bowl, stir together the confectioners' sugar, lemon juice, and milk until smooth. Use a pastry brush to gently brush the icing on each cookie; don't brush icing on the eggs. Place the cookies on a wire rack over a baking sheet and dust with the colored sprinkles. Add the sprinkles while the icing is still wet so that they will adhere.
Allow the icing to dry and harden before storing. The cookies may be stored in a covered container in the refrigerator for about a week.