In a large mixing bowl, whisk together the oat flour, coconut flour, and sea salt. Add the coconut oil, maple syrup, and vanilla, and stir well to combine. Let the dough rest for 5 minutes to thicken, stirring once or twice. You should have a soft, slightly spongy dough.
Gather the dough and form it into a log. Wrap in plastic wrap and chill for 30 minutes to 1 hour. You want a happy medium here—the dough should be chilled enough to roll out smoothly (instead of ripping/tearing) but not so chilled that it crumbles when you roll it out. If you over-chill the dough, simply knead it between your hands for a few minutes to gently warm it.
About 10 minutes before the dough has finished chilling, preheat the oven to 350F. Line two large cookie sheets with parchment paper.
For cookie cutter cookies: Unwrap the dough and cut it into two pieces. Knead each piece a bit to warm it to the right texture (should feel like chilled play-doh). Lightly dust a clean workspace with oat flour and use a rolling pin to roll one of the pieces of dough into a 1/4-inch-thick oval (it's okay if the ends fray or tear a bit—that's normal for shortbread dough). Use your cookie cutters of choice to punch out pieces of dough, and carefully transfer each to one of the lined cookie sheets. Repeat the process with the remaining dough.
For simple round cookies: Slice the log into 1/4-inch-thick circles. Transfer to the lined cookie sheets.
Bake the cookies for 12 to 14 minutes, or until just beginning to turn golden at the edges. Cool completely on the pans before frosting.
Meanwhile, prepare the royal icing.
Once the cookies have cooled, frost them as desired. Allow the icing to dry for 1 hour, or until it's well set.
Store in an airtight glass container for up to 1 week.