Beat softened butter in bowl of a stand mixer fitted with the paddle attachment on medium-high speed until creamy. Add horseradish, thyme, lemon zest and juice, and sea salt. Beat on high speed, stopping to scrape down the sides of the bowl once or twice, 1 minute.
Preheat oven to 450°F. Line a rimmed baking sheet with rock salt. Place oysters in a single layer on rock salt.
Top each oyster with 1 rounded teaspoon of butter mixture. Combine breadcrumbs and oil in a small bowl, and toss to coat; sprinkle over oysters.
Roast in preheated oven until edges of oysters start to pull away and curl, and breadcrumbs are browned and crispy, 15 to 20 minutes. Serve immediately.
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