Preheat oven to 375 degrees F. Coat a 9x13-inch baking dish with olive oil; set aside.
In a large skillet over medium heat, warm 2 tablespoons olive oil. Add onions and cook until soft and translucent, about 5 minutes. Add a pinch chili flakes and sausage and cook, stirring occasionally to break into pieces, until sausage is cooked through and browned.
Add 2 tablespoons butter to the skillet to melt, then stir in the flour until completely combined. Stir in chicken stock and milk, increase heat to bring to a boil, and cook until thickened, about 5 minutes. Season to taste with salt and pepper, then remove from heat and set aside to cool.
Heat 2 tablespoons butter in a large nonstick skillet over medium heat. Add eggs and cook, stirring often, to scramble. Remove from heat and set aside.
To assemble lasagna, spoon 1/2 cup gravy, without sausage pieces, into the bottom of the baking dish and spread to coat. Place 4 lasagna noodles over the bottom of the pan, then top with half of the eggs, half of the spinach, 1/3 of the sausage gravy, 1/2 cup shredded cheddar, 1/2 cup shredded mozzarella and 1/3 of the hashbrowns. Repeat with another layer of noodles, the remaining eggs, remaining spinach, sausage gravy, 1/2 cup shredded cheddar and 1/2 cup shredded mozzarella. Top final layer of noodles with remaining sauce, cheeses and hash browns.
Cover and bake 30 minutes, the uncover and bake 10 minutes more. Let rest 10 minutes before slicing. Garnish with chopped chives to serve.