Place the butter and sugar in a medium bowl and cream together by hand until smooth and combined.
Stir in the vanilla extract.
Add the flour and salt and mix until the dough starts to come together.
Add the milk and mix until you have a smooth, cohesive dough.
Roll the dough between two sheets of parchment paper to about 4 mm thick. Chill the rolled dough for 15 minutes.
Cut the dough into rounds using a 2 cm cutter, and transfer to a sheet pan lined with parchment paper. Chill for another 10 minutes.
While the dough chills, preheat the oven to 350°F conventional.
Bake for 11 minutes, until set and only lightly golden around the edges.
Leave the biscuits on the sheet pan for about 5 minutes, then transfer to a wire rack and cool completely.
To make the royal icing
Whisk the egg white and lemon juice in a clean bowl until lightly frothy.
Gradually add the sifted powdered sugar, whisking well until the icing becomes thick, glossy and very stiff.
The icing is ready when it forms stiff peaks that hold their shape without drooping. If it's not stiff, add more powdered sugar.
Divide the icing between four small bowls and color with gel food coloring (as desired).
Transfer each color to a piping bag fitted with a star nozzle.
Pipe a small star of icing onto each cooled biscuit - dividing the colors evenly between the biscuits.
Leave the Iced Gems uncovered at room temperature overnight, until the royal icing is completely hard and crunchy.
Notes
Royal icing: The icing needs to be very stiff. If it is too soft, add a little more sifted powdered sugar.Food coloring: Gel food coloring works best because it adds color without significantly thinning the icing.Setting: Leave the finished Iced Gems uncovered overnight until the royal icing is completely hard before storing.Storage: Once the icing is completely dry, store the Iced Gems in an airtight container at room temperature.