It's do-able sure....but a lot of work and each step must be done very quickly before the icing starts to dry/crust or it won't look good at all.
Here's how I would do it.
-make up the thinned royal icing recipe
-split recipe into two bowls
-tint one bowl blue, tint one bowl brown
-put each icing colour in a piping bag
-outline all the cookies using either brown or blue (about a #1 tip)
-when dry (10 minutes), take one colour and fill in half the cookie then immediately take the other colour and fill in the other side of cookie (about a #2 tip)
-using a pin or a skewer, swirl the middle of the cookie to create a mixing of both colours (see my Autumn Leaf cookie photo for example of this technique)
-shake the cookie softly to settle the icing and quickly fill in any gaps
-take the opposing icing colour and write the initials on either side (this MUST be done before the icing starts to crust, so that the icing literally sinks into the icing and is flat)
-do not shake the cookie after this step or you risk distorting the lettering