Using a hand mixer or a stand mixer fitted with paddle attachment, beat the butter and the sugar over medium-high speed until they are combined. Add the vanilla and mix well. In a medium bowl, sift together the flour and the salt.
Add the sifted flour mixture to the butter/sugar mixture and mix on low speed until the dough starts to come together. Dump the dough onto a lightly floured surface. The dough will be very crumbly. Shape the dough into a disk. Wrap in plastic and chill for 20 minutes.
Preheat the oven to 350 degrees Fahrenheit.
Roll the dough 1/4 inch thick and cut into bunny shapes with a cookie cutter. Place the cookies on a baking sheet lined with parchment paper or on a baking mat. I like to chill the cookies for 5-10 minutes before baking however this is NOT a necessary step. See notes.
Bake for 20 - 22 minutes or until the edges begin to brown. Allow to cool to room temperature.
When the cookies are cool, place them on a baking sheet lined with parchment paper. Place the shredded coconut on a plate.
Put 2-3 ounces of white chocolate in a small glass bowl and melt in the microwave in 30 second increments, stopping and stirring after each until completely smooth. Use this to dip the marshmallows.
Melt the remaining white chocolate or the candy melts, if using, until smooth (please read the notes). Dip the bottom half of the bunny cookies into the melted chocolate or candy melts. Place the dipped cookies on a cooling rack or baking sheet lined with parchment paper.
With the help of a fork, dip the mini marshmallows into the melted white chocolate and immediately roll it into the shredded coconut. Place the fluffy tails (coconut covered marshmallow) on a plate for about 4-5 minutes. Grab each fluffy tail carefully and place them on top of the dipped cookie (in the center). If the melted chocolate on the cookie has dried up, simply dip the bottom of the coconut covered marshmallow in white chocolate (as if using glue) and place on cookie.
Let the cookies dry up for about 10 minutes.