Pour oats into a food processor or blender and grind until they have a smooth texture. Whisk together the oats, flour, cinnamon, nutmeg, baking soda and salt in a large bowl.
Beat butter, brown sugar, sugar and vanilla in a large mixer bowl until creamy. Add the eggs and beat well. Gradually add the flour mixture until just blended. Add carrots and mix just until combined; dough will be sticky.
Drop 1-inch balls of dough onto greased or parchment lined baking sheets. Bake for 12 to 15 minutes until cookies are set and just browning around the edges. Cool 5 minutes on baking sheets before removing cookies to wire rack to cool completely.
Beat all filling ingredients with electric mixer in large bowl, adding enough lemon juice to make a creamy consistency. Spread filling over bottom of one cookie and then press another cookie, bottom-side down into filling. Refrigerate cookies if not serving immediately.