Preheat the oven to 350 degrees F. Butter a 2-quart baking dish or gratin dish.
Melt 2 tablespoons butter in a medium saucepan over medium-high heat. Add the onions and 1 teaspoon salt and cook, stirring frequently, until the onions are translucent, about 4 minutes. Stir in the flour and stir to coat the onions. Cook for 2 minutes. Remove from the heat and whisk in the hot milk. Return to the heat and bring to a simmer. Stir in the cheese by the handful. Add salt to taste. Add the broccoli to the cheese sauce and pour into the prepared dish.
Melt the remaining 2 tablespoons butter in a small skillet and stir in the cracker crumbs. Cover the broccoli casserole with the buttered cracker crumbs. Bake until bubbly and lightly browned, about 20 minutes.
To steam, put the broccoli in a microwave-safe bowl. Add a little water to the bottom of the bowl and top with a microwave-safe plate. Microwave on high for 2 to 3 minutes, until bright green and crisp-tender. Drain.