Mix the sweetened condensed milk with water and set aside.
In the bowl of an electric mixer, mix the flour, baking powder, sugar, cardamom, and salt on low speed until well mixed.
Add the butter and mix until the mixture resembles cornmeal, about 5 minutes.
Mix in the condensed milk mixture all at once and continue mixing until it is fully incorporated and the dough forms a ball around the paddle.
Transfer to a large sheet of plastic wrap, pat into a 1-inch-thick disk, and wrap tightly in the plastic.
Refrigerate for 20 minutes.
Preheat the oven to 300 F.
Break off a ball of dough with the diameter of a dime (a tiny piece). Roll the dough into a five inch long snake. Try to make the snake fairly thin, but with enough thickness that you can still pick it up and move it without it breaking.
Transfer the snake to a parchment- or Silpat-lined baking sheet (or just roll it directly on the sheet if that's easier for you) and repeat with the remaining dough, setting the sticks 1/2 inch apart.
Refrigerate the baking sheets, uncovered, for 10 minutes.
Brush the dough snakes with the egg white and bake until golden brown and crisp. Check on them at 20 minutes, but depending on how thick you made them, it may take 25 minutes.
Melt white chocolate in a double boiler or using short intervals in the microwave. Stir until smooth.
Divide melted chocolate into three tall shot glasses.
Add red food coloring to one of the shot glasses. If the chocolate is no longer smooth and liquidy, add a little vegetable oil at a time to the red chocolate until the consistency is good for dipping.
Add blue food coloring to another one of the shot glasses. If the chocolate is no longer smooth and liquidy, add a little vegetable oil at a time to the red chocolate until the consistency is good for dipping.
One at a time, dip 1/3 of the Pocky sticks in the plain white chocolate. Set each one back on the lined cookie sheet when done.
Repeat with the other two colors.
Refrigerate until just before serving to harden the chocolate.