Add 1 teaspoon of oil to the pan along with the shallots and a pinch of sea salt. Cook for about 10 minutes, stirring frequently, until the shallots have begun to caramelize. Remove from the pan and set aside.
Turn the heat down to medium and add a tablespoon of oil to the pan. Place the sprouts cut side down in the oil. (You may need to do this in two batches depending on the sizes of your pan.) Sprinkle with a pinch of salt Leave the sprouts untouched for about 5 or 10 minutes. Check the bottoms to see if they've begun to brown.
Sprinkle the sprouts with salt and pepper and toss to coat, and cook for another five minutes, stirring frequently. Add the shallots and pistachios to the pan and cook for another minute or two, until heated throughout.