Melt butter in a small saucepan over low-medium heat, stirring frequently, until it foams, the foam dissipates, it turns a deep golden brown, and develops brown flecks and a nutty fragrance, about 10 minutes or so. Toward the later stages, around the time it begins to foam, watch the brown butter very closely. It can go from 'brown' to 'burned' very quickly.
Remove from heat, and cool for 10 minutes.
Carefully transfer to a heatproof jar or bowl, making sure to get alllllll the particles and brown bits from the pan... these are important!
Refrigerate brown butter until firm.
Bring brown butter to room temperature... I know, I know.
In a large mixing bowl, beat the brown butter, oil, and sugars until combined.
Beat in eggs and vanilla.
In a large bowl, whisk together flour, baking soda, cream of tartar, and salt.
In two additions, add flour mixture to the butter mixture, beating until just combined. Do not overmix. Dough will be thick and a little crumbly.
Line baking sheets with parchment paper or silicone baking mats.
Drop dough by rounded 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 completely.