Beat eggs in a bowl. Whisk in milk, heavy cream, 1/3 cup sugar, vanilla extract, nutmeg, and salt.
Working in batches, dip 10 slices day-old brioche into batter to coat. Overlap brioche slices in a 9-by-13-inch ceramic or glass baking dish. Pour remaining batter over top. Refrigerate, covered, for at least 2 hours.
Preheat oven to 375 degress. Top with pecans and sprinkle with 3 tablespoons sugar. Bake, covered with parchment-lined foil, for 25 minutes. Uncover; bake until top is golden brown and crunchy, 20 to 25 minutes more. Serve with syrup.