On a large rimmed baking sheet, toss the shallots with 1/4 cup of the flour; shake off any excess flour.  In a large, deep skillet, heat 1 inch of oil until shimmering. Add the shallots in 2 batches and fry over moderate heat until very crisp, about 2 to 3 minutes.  Using a slotted spoon, transfer to paper towels, then sprinkle with salt.

In a large pot of boiling salted water, cook the beans until just tender, about 5 minutes.  Drain and refresh under cold running water; drain and pat dry.

Melt the butter in a large heavy-duty skillet.  Add the onion and cook over low heat, stirring occasionally, until softened about 5 minutes.  Add the paprika, cayenne (if using) and a pinch of freshly ground black pepper; cook for 1 minute, stirring.  Add the mushrooms, cover and cook over moderate heat until softened about 5 minutes.  Uncover and cook, stirring, until browned, about 5 minutes longer.  Stir in the remaining 1 1/2 tablespoons of flour and gradually stir in the stock until smooth.

Simmer the mushroom sauce over low heat, stirring, until thickened, about 3 to 5 minutes.  Stir in the crème fraîche, lemon juice, and beans.  Cover and simmer, stirring occasionally until the beans are heated through, about 5 minutes.  Season with salt and pepper and transfer to a large glass or ceramic baking dish.

Cover the casserole with foil and bake until bubbling, about 20 minutes.  Uncover, scatter the crispy shallots on top and serve.