In bowl of a stand mixer or by hand, cream softened butter (or vegan margarine) with powdered sugar until light and fluffy - approx. 5 minutes. Add vanilla and beat for another minute or so until well incorporated.
In a bowl, whisk together flour, matcha and salt until well incorporated. Sift flour mixture over the creamed butter mixture and mix until all flour is combined with the butter and the dough has come together nicely.
Preheat oven to 325f degrees. Gently knead the dough and shape into a flattened disc. Roll out the dough between two sheets of parchment paper and roll into a rectangle (give or take) until it is uniformly about 1/4" - 3/8" thick.
Cut out gingerbread men shapes (or any other shape) and place each cookie on a silpat or parchment lined baking sheet. After you've cut as many shapes as you can, gather the dough scraps together, gently knead them into a ball, shape into a flattened disc and re-roll to 1/4"-3/8" thick, cut out cookies and continue this until all of your dough is used up.
Bake for 15 minutes and then let cookies completely cool on the cookie sheet. Once cookies are cool, set up a double boiler over low heat and melt about 4 ounces of white chocolate, stirring continuously over the heat until just melted.
Remove white chocolate from double boiler, stir until perfectly smooth and then scoop white chocolate into a piping bag fitted with a #2 icing tip.
If you are decorating little men, start by first outlining the edges of each cookie. Decorate each cookie as desired.
Small dots, like the eyes and the buttons may have a slight point when you pull the icing tip away from the cookie - let the white chocolate set on the cookie for about 10 minutes and then with a barely wet finger tip, smooth away the slight points.