Preheat oven to 350°. Whisk together the first 10 ingredients until blended. Gently stir in bread. Transfer to a greased 3-qt. or 13x9-in. baking dish. Sprinkle with brown sugar; let stand until bread is softened, about 15 minutes. Bake until puffed and a knife inserted in center comes out clean, 40-45 minutes., Meanwhile, for sauce, melt butter in a large heavy saucepan over medium heat. Add brown sugar; stir until dissolved. Gradually stir in cream. Bring to a boil. Reduce heat; simmer until thickened, 15-20 minutes, stirring constantly. Remove from heat; add vanilla, sea salt and baking soda, stirring well. Serve sauce with warm bread pudding.