Preheat the oven to 350 degrees. Lightly butter 2 6-cavity doughnut pans.
In a large bowl, whisk together flour, sugar, baking powder, cinnamon, nutmeg, and salt.
In a blender, puree raspberries and then strain.
In a smaller bowl, whisk together the egg, milk, melted butter, vanilla, and ¼ cup raspberry puree. Reserve remaining raspberry puree for the glaze.
Combine the wet mixture into the dry ingredients and mix until just combined.
Spoon the batter into the baking pans, filling each one about three-quarters full. Bake for 12 minutes. Allow to cool in the pan for 5 minutes, then place doughnuts onto a cooling tray.
Melt butter in a sauté pan over low heat. Add sugar, vanilla, milk, and ¼ cup raspberry puree and whisk until smooth. Remove from heat and let cool.
Once donuts have cooled, dip each top in the glaze and then return to cooling rack to let glaze harden.