Heat the olive oil in a large skillet over high heat. Add the mushrooms and cook until golden, about 4 minutes. Reduce the heat to medium; add 1/2 teaspoon salt, pepper to taste, the celery and shallot and cook until the vegetables are tender, about 4 minutes. Add 6 tablespoons flour, the butter and nutmeg and cook, stirring, until the flour is toasted, 1 to 2 minutes. Slowly whisk in the broth and cook, stirring, until the mixture is the consistency of gravy, about 4 minutes. Slowly stir in the milk and simmer, stirring, until very thick, 6 to 10 more minutes. Remove from the heat, transfer to a bowl and let cool to room temperature. Cover and refrigerate until firm, at least 2 hours.
Line a baking sheet with parchment paper. Pulse the bread in a food processor to make fine crumbs; transfer to a shallow dish. Put the remaining 1 1/4 cups flour in another shallow dish; beat the eggs in a third dish. Drop tablespoonfuls of the chilled mushroom mixture into the flour and toss to coat, then shape into 1 1/4-inch-long croquettes. Dip in the eggs, then roll in the breadcrumbs and transfer to the prepared baking sheet. Cover with plastic wrap; refrigerate until firm, 30 minutes.
Heat 1/2 inch vegetable oil in a deep saucepan until a deep-fry thermometer registers 360 degrees F. Working in batches, add the croquettes and fry until golden, 2 to 3 minutes per side. Remove with a slotted spoon and drain on paper towels; season with salt. Serve with mustard.