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Preheat oven to 400 degrees. 4 individual shallow baking dishes.
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Drain oysters into a large bowl, reserving liquor. Check oysters for any pieces of shell and discard. Strain liquor through a fine mesh sieve; set aside.
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Melt butter in a small saucepan over low heat. Whisk in flour, then add reserved oyster liquor and broth or milk. Bring mixture to a boil, whisking constantly. Whisk in cream and reduce heat to a simmer. Cook for 10 minutes, stirring occasionally. Season with salt, pepper, hot sauce, and nutmeg. Remove mixture from heat; cover sauce to keep warm.
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Heat olive oil in a medium skillet over medium-high heat. Add eggplant and cook, stirring occasionally, until browned, about 5 minutes. Add onion, tasso, and garlic; stir in sage and rosemary. Cook until eggplant is tender and cooked through, 5 to 7 minutes more. Season with salt and pepper; transfer mixture to a colander; let stand to drain excess oil, about 5 minutes.
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Spread mixture evenly in baking dishes; top with an even layer of oysters. Drizzle with sauce and sprinkle over breadcrumb topping. Transfer to oven and bake until golden brown and bubbly, about 10 minutes. Serve immediately.