Preheat oven to 375°F with rack in middle. Butter a 2-qt shallow baking dish.
Whisk together milk, cream, eggs, and 1/4 teaspoon each of salt and pepper in a large bowl.
Cook bacon in a 12-inch heavy skillet over medium heat, turning occasionally, until crisp. Transfer with tongs to paper towels to drain, then coarsely crumble.
Pour off all but 1 tablespoon fat from skillet. Increase heat to medium-high and cook shallot and garlic, stirring constantly, until golden, about 1 minute. Gradually add arugula and cook, stirring, until it wilts.
Stir arugula mixture, bacon, bread, and cheese into custard. Transfer to baking dish and cover with foil. Bake 30 minutes, then remove foil and bake until golden in spots, about 10 minutes more.