Preheat your oven to 350 degrees, making sure that the rack is in the upper third of your oven.
In a small saucepan combine the milk, cream, garlic clove and sprig of thyme. Bring the mixture just to a simmer then remove from the heat and let the ingredients steep while you prepare the rest of the dish.
In a large skillet, heat 1 tablespoon of oil over medium high heat and add the Swiss Chard or spinach to the pan. It is going to seem like a lot of greens, but this will all cook down to a much more manageable mass. Add ½ cup of water to the hot pan and cover with a lid. Cook the greens until the stems are softened and the leaves are wilted, about 5 minutes. Meanwhile, bring a pot of salted water to a boil and add the broccoli to the boiling water. Blanch the broccoli for about 5 minutes, then drain and rinse with cold water to stop the cooking.
Put the broccoli and Swiss chard in the bottom of a 2-quart baking dish and sprinkle with ½ t. salt and ½ t. pepper. Then, strain the cream mixture into a mixing bowl and whisk in the eggs, remaining salt and pepper, and Parmesan cheese. Pour the egg mixture over the vegetables. Stir the remaining one tablespoon of oil together with the bread crumbs and a pinch of salt and pepper, and then sprinkle over the top of the gratin.
Bake until the custard is set, about 30 minutes, then turn on the broiler and broil until the bread crumbs are golden and crisp, 2 to 3 minutes. Let the gratin stand for 5-10 minutes before serving.