Line two sheet pans with parchment. Set a wire rack over one lined pan. Chill the cookies on a plate in the fridge for 10 minutes to help the coating set with a clean snap.
Melt the dark chocolate: Build a double boiler by setting a heatproof bowl over a pot with 1 inch of barely simmering water (bowl should not touch water). Add the semisweet chocolate, 1 1/2 teaspoons oil, and the pinch of fine sea salt. Stir constantly until smooth and glossy, 3–4 minutes, then remove the bowl from the pot and wipe any condensation from its bottom.
Dip the cookies: Working with one chilled Oreo at a time, submerge it in the melted chocolate. Lift with a fork, tap the fork gently on the bowl’s edge to shed excess, and scrape the underside of the fork on the rim. Set the coated cookie on the wire rack to drip for 1–2 minutes.
Transfer each cookie from the rack to the parchment-lined pan to finish setting without pooling. Repeat with remaining cookies. Tip: If you notice tiny air bubbles on the surface, pop them with a toothpick before the chocolate firms up.
Make the drizzle: Place the white chocolate and remaining 1/2 teaspoon oil in a microwave-safe bowl. Heat at 50–60% power in 20-second bursts, stirring well between each, until just melted and smooth. Do not overheat.
Drizzle and decorate: Using a small spoon, flick thin ribbons of white chocolate over the coated cookies. While the drizzle is tacky, add sprinkles or a tiny extra pinch of sea salt if using.
Set and serve: Refrigerate the tray for 10 minutes to set completely, then bring to room temperature for 5 minutes before serving. If the coating thickens as you work, briefly rewarm the bowl over the warm pot (or 5–10 seconds in the microwave) to loosen.