Butter a 9x13-inch shallow baking dish (with handles, if possible) with the 2 teaspoons softened butter.

Set a large skillet (preferably nonstick) over high heat with 1 tablespoon of the butter. When the butter froths, add just enough apple slices to cook in one layer; saute until browned on both sides, about 5 minutes. Transfer to a large plate and saute the remaining apple slices the same way, adding 1 tablespoon more butter for each batch. The second (and third) batch will brown more quickly, so you may need to lower the heat. Reduce the heat to low and melt 4 tablespoons butter in the empty pan, cooking it and swirling the pan until the butter turns light brown, about 1 minute. Take it off the heat immediately.

Lay one-third of the bread in the buttered baking pan, scatter about one third of the sauteed apples and drizzle on one-third (about 1 tablespoon) of the brown butter. Make a second layer of bread, apples, and brown butter, and finish with a layer of bread and brown butter; reserve the remaining apples.

Pour the milk and cream into a medium saucepan. Stir in the 3/4 cup brown sugar and salt. Heat over medium high, stirring once or twice, until the sugar dissolves and the milk is hot; don't worry if it looks like it's curdling.

In a large mixing bowl, whisk together the eggs, yolks, and vanilla. Gradually pour about 1 cup of the hot milk mixture into the eggs, whisking constantly, and then whisk in the rest. Gradually whisk in the sour cream to combine completely.

Pour the custard over the bread, pressing the bread down to fully moisten it. Lay plastic wrap on the surface of the pudding and let it sit, pressing on the plastic frequently to submerge the bread, until the bread is thoroughly soaked, 30 to 40 minutes.

While the bread soaks, adjust an oven rack to the center position. Set a roasting pan, large enough to comfortably hold the pudding, on the rack. Pour in hot water to come one-quarter of the way up the sides. Heat the oven to 350 degrees F.

Remove the plastic wrap. Arrange the reserved apples on top and sprinkle the top evenly with 1/4-cup brown sugar (or more, if you like). Set the pudding in the roasting pan; if the hot water doesn't reach halfway up the sides, add more until it does. Bake until a wooden skewer inserted in the center comes out clean, 40 to 50 minutes. Remove from the water bath and turn on the broiler. Broil the pudding (the top should be about about 3 inches from the heat source) until it's lightly caramelized, 2 to 4 minutes. Serve warm.