Bring a pot of water, large enough to cover the broccoli, to boil with the kosher salt. Have a large bowl of ice water nearby. I don't chop the broccoli before I boil it because I do not enjoy the mess in the pot, but to each his own. Boil the broccoli until just tender - do not overboil. Mine took about 5 minutes. Stop the cooking and discoloration by lifting the broccoli with tongs and placing in the ice water.
Turn your burner to medium or medium low, pour out the water and add the butter and olive oil. Add the shallot and cook until beginning to caramelize, about 5 minutes, stirring constantly. Adjust heat to prevent scorching. Set aside.
Drain the broccoli and roughly chop. Add the broccoli, cooked shallot, cheese, sour cream, and 1/2 cup whole milk to a glass blender. Begin to puree, and add more milk until it's a consistency you like. Mine was very thick and creamy. Some people may like it soupier. Salt and pepper to taste. If you're anything like me, you'll justify needing to taste it repeatedly before anyone else gets any.
It will be lukewarm or cool at this point. So you can either serve as is, chill it, or warm it up in the pot you cooked the broccoli and shallot. Either way, it is rich and delicious. Also, you may like to thin it out with chicken stock after the first 1/2 cup of milk if that suits your taste or needs better.