In a large mixing bowl, beat the butter, oil, and sugars until combined.

Beat in eggs, vanilla, and rum extract.

In a large bowl, whisk together flour, baking soda, cream of tartar, and nutmeg.

Gradually add flour mixture to the butter mixture, beating until just combined after each addition. Do not overmix.

Line baking sheets with parchment paper or silicone baking mats.

Drop dough by teaspoonfuls onto baking sheets.

Bake at 375 for 8-11 minutes, or until edges and bottoms are lightly browned.

Remove to wire racks to cool.

Whisk together butter, vanilla, rum extract, 1/4 cup milk, and nutmeg, until smooth. 

Stir in 4 cups powdered sugar, mixing until smooth.

Add additional milk or powdered sugar, as needed, to achieve a thick, but spreadable consistency. It should be thicker than a glaze, and it shouldn't run off the cookie, but should be easily spreadable. 

Frost cookies, dust with additional nutmeg, if desired, and allow to set.