Using an electric mixer (stand or hand-held), cream together butter and sugar until pale and creamy, then beat in whole egg. Scrape down sides of bowl. Beat in milk, almond and vanilla extracts and salt. Gradually add flour and ground almonds, mixing until a soft dough forms (you may need to mix in the last additions of flour by hand).
Divide dough into 3 equal portions, each in its own bowl. Leave one plain, tint the second with green food colouring and the third with red. Mix in colourings thoroughly. Cover each dough with plastic wrap and chill until firm and no longer sticky, about 45 min.
Cut 3 pieces of parchment paper at least 10 x 12 in. (25 x 30 cm) in size. Using a floured rolling pin, roll each dough into a 7 x 9-in. (18 x 23 cm) rectangle, approx. 1/8 in. (3 mm) thick on its own piece of parchment. Patch edges as required to attain size. Chill dough, still on parchment, on baking sheets until firm.
In a small bowl, beat egg white. To layer, place red dough (still on parchment) on work surface and brush with egg white. Lining up the edges, invert plain dough onto red one and peel off parchment from plain dough. Brush with egg white and invert green dough on top, again peeling away parchment. Use a sharp knife to lightly trim edges, wiping blade clean between cuts to keep colours distinct.
Place longest side of dough nearest to you and tightly roll up into a log, smoothing and straightening as you roll (lift parchment to help release red dough). Gently press edge of log together to seal. If log’s diameter is thicker than approx. 2 in. (5 cm), gently hand-roll to lengthen. Wrap log in plastic wrap and chill until firm, approx. 2 hr. (or overnight).
Preheat oven to 350°F (180°C). Slice log into 1/4-in. (5 mm) thick rounds, wiping blade clean between cuts. Set about 1 in. (2.5 cm) apart on 2 parchment paper-lined baking sheets. Bake 12 to 14 min., until firm but not browned. Cool completely before storing in an airtight container.