For the French toast and caramelized apples: Preheat the oven to 275 degrees F.
Spread the bread cubes on a large baking sheet and bake until lightly toasted, about 15 minutes. Let cool. Adjust the oven to 350 degrees F.
Combine the sugar, apple cider and lemon juice in a large, high-sided saute pan and bring to a simmer over medium heat. Stir in the Gala and Granny Smith apples and cook, stirring occasionally, until softened and caramelized, about 15 minutes. Add the apple brandy if using; cook a few minutes longer. Using a slotted spoon, transfer the apples to a bowl (you don't want the apples to be too wet) and let cool. Reserve the apple cooking liquid for serving.
For the streusel: Mix together the butter, salt and sugar in a large bowl until smooth. Add the flour, oats and cinnamon and mix until well combined. Refrigerate for 15 minutes. Squeeze the mixture in your hands to form larger and smaller clumps.
For the casserole: Butter a 9-by-13-inch baking dish with some softened butter. Whisk the eggs, yolks, vanilla and cinnamon in a large bowl until smooth. Whisk in the half-and-half, then fold in the cooled apples. Add the bread and stir to coat, pressing down to make sure bread is totally submerged. Let sit for 10 minutes.
Transfer the apple-bread mixture to the buttered baking dish, pressing down to make sure the top is an even layer. Scatter the streusel evenly over the top. Cover with foil and bake for 30 minutes. Uncover and continue baking until puffed and lightly golden brown, another 30 minutes or so. Remove from the oven and let cool slightly.
Reheat the reserved apple cooking liquid. (If very thin, simmer briefly over medium heat until reduced to a syrup; if there isn't much, add some apple cider and simmer briefly.) Turn the heat to low and whisk in the cold butter, piece by piece, until emulsified. Serve with the French toast.