Preheat oven to 350 degrees. Spray an 8×8 baking dish with nonstick spray then place on top of a sheet pan (in case of spillovers) and set aside.
Bring 2-3/4 cups chicken broth to a boil in a medium-sized saucepan then add quinoa, place a lid on top, turn heat down to low, and simmer until nearly all the liquid has been absorbed and quinoa is tender, 15 minutes. Add broccoli then place the lid back on top, remove from heat, and let steam for 10 minutes.
Meanwhile, melt 2 Tablespoons butter in a large skillet over medium-high heat. Add mushrooms and shallots, season with salt and pepper, then saute until mushrooms are golden brown. Add garlic then saute for one more minute.
Add remaining 2 Tablespoons butter then sprinkle in flour when melted and saute for one minute. Slowly stream in remaining 1 cup chicken broth and milk while whisking to avoid lumps. Season generously with salt and pepper then bring mixture to a simmer while stirring constantly. Turn heat down to medium then simmer until thick and bubbly, 5 minutes.
Remove from heat then stir in the broccoli , quinoa, and half the cheese until smooth. Scoop the mixture into prepared baking dish then sprinkle remaining cheese on top then bake for 15-20 minutes, or until cheese is golden brown and bubbly. Let sit for 10 minutes before scooping and serving.