Preheat the oven to 350 degrees F (175 degrees C).
Melt 1 stick butter in a large frying pan over medium heat. Add onion and garlic and saute until onion is translucent, about 5 minutes.
Add flour and stir quickly until it is mixed like Play-Doh®. Quickly add broth, milk, and chicken bouillon; whisk until smooth. Remove from the heat and transfer to a large bowl. Add sour cream and both cheeses; stir until well mixed.
Place shredded chicken across the bottom of a 9x13-inch casserole dish. Layer cooked broccoli over the chicken and season with salt and pepper. Pour the cheese mixture over top.
Bake, uncovered, in the preheated oven until hot, about 25 minutes.
Meanwhile, place crushed cornflakes in a zip-top bag. Add melted butter and seal the bag. Massage with your hands until butter coats all the cornflakes.
Remove casserole from the oven and distribute cornflake mixture evenly over the top. Bake for 5 more minutes.