A quick and dirt trick I came up with
Cut the letters out separately from the background. Using Profile cuts this has the advantage of going much faster. I used a v bit to carve a thin outline of the letters into the back piece as a placement guide, but if it is simple enough you might not need it.
Paint the letters - if using spray paint I recommend the Rustolium 2x painters choice - cheep paint just gets sucked in and takes way to many coats to seal the MDF.
You might want to sand the letters after painting as the cut edges will plump a bit, then give a light second coat.
Then flip the letters over face down, make sure they are on a flat surface so that none of the top is exposed. Then spray paint the back and sides in your shadow color.
(I put mine on a sheet of wax paper as I was worried about the still possibly tacky top coat sticking)
Glue the letters in place on the sign back.
Quick and easy.
Or you could try hand painting the edges with a brush - whatever you are more comfortable with.
I used this trick on my MDF Pumpkins to paint the edge of the back part orange to match the edge of the front.