Preheat the oven to 425 degrees F (220 degrees C).
Whisk flour, oats, baking powder, salt, 1 teaspoon cinnamon, nutmeg, 1/2 teaspoon ground ginger, and cloves together in a mixing bowl. Cut cold butter into mixture with a pastry blender or fork until the mixture resembles coarse crumbs the size of peas or smaller. Whisk pumpkin purée, 2 tablespoons cold milk, egg, and vanilla together in a small bowl; fold into flour mixture until just blended.
Divide dough into two sections on a lightly floured surface. Gather each into a ball and knead about 10-12 times. Flatten balls to 1/2-inch thickness on a nonstick baking sheet. Leave about 2 inches between rounds. Cut rounds into sixths with a butter knife, but do not separate wedges.
Bake in the preheated oven until the bottoms and edges are just golden brown, 12 to 16 minutes. Set aside to cool.
While scones cool, mix confectioners' sugar, 1 tablespoon milk, remaining pinch of ginger, and dash of cinnamon together in a small bowl. Add more milk to thin frosting as needed. Pour frosting into a resealable plastic bag. Snip off a corner of the bag and drizzle icing over the cooled scones in a zig-zag fashion. The icing will harden as it cools.