Melt the butter with the olive oil in a large frying pan over medium high heat. Add the onions and stir to coat i the mixture, then reduce heat to low. Continue cooking until the onions are a deep gold and smell slightly sweet, about 1 hour, stirring every 5 to 10 minutes. Remove from heat and set aside.

Preheat the oven to 425 degrees Fahrenheit. Place the sweet potatoes on a pan lined with tin foil and bake until tender, about 1 hour. Remove from the oven but leave the oven on.

Spread out the pecans on a baking sheet and roast in the oven for 5 minutes. Remove the pan from the oven and turn it off. Allow the sweet potatoes and pecans to cool.

Meanwhile, melt the butter in a small saucepan over medium low heat, stirring occasionally. Continue cooking the butter, stirring several times to ensure it cooks evenly, until the liquid turns deep gold and the mild solid bits at the bottom of the pan are dark brown and the mixture smells toasty and nutty. Remove from heat and set aside.

Once the sweet potatoes have cooled enough to handle, peel off and discard their skins using your hands. Place the sweet potatoes, pecans, brown butter, milk, salt, and sage in a blender or food processor and puree at medium low speed until smooth, stopping to scrape down the sides as necessary to ensure a thoroughly blended mixture.

Empty the puree into a mixing bowl and stir in the caramelized onions. Serve immediately.