Preheat oven to 400℉.
Arrange bread on rimmed baking sheet.
Bake until pale golden, about 5 minutes, then cool.
Butter an 11x7-inch glass baking dish.
Melt 2 tablespoons butter with oil in a heavy large skillet over medium-low heat.
Add potatoes.
Stir to coat and arrange in a single layer.
Cover and cook until potatoes are almost tender, about 8 minutes.
Uncover; increase heat to medium-high, and cook until potatoes are lightly browned and tender, stirring occasionally, about 5 minutes.
Add shallots and saute until soft, about 2 minutes.
Remove from heat.
Gently mix in bread, salmon, chives, and minced dill.
Transfer mixture to prepared dish.
Whisk eggs and next 5 ingredients in medium bowl to blend well.
Pour custard over potato mixture in dish.
Let stand 15 minutes, occasionally pressing bread into custard. (Can be made 1 day ahead. Cover; chill. Let stand at room temperature 30 minutes before baking.)
Preheat oven to 350℉.
Bake uncovered, until custard is set, about 30 minutes.
Cut into squares and top with additional sour cream, capers, and dill sprigs; serve hot.