Place the quinoa in fine sieve or strainer and rinse thoroughly under cool running water and allow to drain.
Position a rack in the middle of the oven and heat to 350ºF.
Bring the quinoa, chicken stock, and 2 teaspoons of the salt to a boil in a medium saucepan over high heat. Reduce the heat to low, stir, cover, and cook for 15 minutes. When 5 minutes remain, add the broccoli right on top of the quinoa and replace the lid.
Heat a large, straight-sided saute pan over high heat and add the mushrooms and 1/3 cup of water. Cook until the mushrooms collapse, 2 to 3 minutes, then add 2 tablespoons of the butter and cook until browned. Remove to a large bowl.
Add the remaining 2 tablespoons of butter to the pan along with the onions and the remaining teaspoon of salt. Reduce the heat to low and sweat the onions until translucent, about 5 minutes. Then stir in the mustard, smoked paprika, black pepper, cayenne, and white pepper, and cook for 1 minute more. Add the half-and-half to the pan and heat until warm.
Whisk the egg in a medium bowl and temper with 1 cup of the hot liquid, continuing to whisk to prevent curdling. Then whisk the egg mixture back into the saute pan and add the cheese, stirring until melted.
Add the quinoa, broccoli, mayonnaise, and the melted cheese mixture to the bowl with the mushrooms and stir to combine.
Pour the quinoa mixture into an ungreased 9-by-13-inch glass baking dish. Bake until set and just beginning to brown, about 45 minutes or until the internal temperature reaches 205ºF to 210ºF. Cool 10 minutes before serving.