You are describing a valve cover gasket that's leaking onto the exhaust.
The engine is raised on the front and tilted to the right (exhaust side) and the oil gets pumped to the top and it flows by gravity back to the oil pan to start the journey again. The oil flows to the back of the engine to the #6 cylinder area. There are drain holes at each cylinder, but any oil that misses the drain holes at the top (#1 cylinder) flows to the next and so on until the oil pools over #6 as it prepares to drain to the oil pan. If the valve cover gasket is a fail, oil will drip onto the exhaust manifold, typically at #4, #5, & #6, or down the back of the engine between the engine and the firewall. All of the stuff that the oil can drip onto is hot and the smell of burning oil is a common complaint. The oil can actually burn off and you do not observe wet oil but you might see the stains that result. Occasionally you can put a turn or two on the valve cover bolts if they have backed out, but replacing the gasket is the thing that smart money would do. When the gasket is replaced, there are 15 or 16 screws that hold the cover on. You must replace the rubber grommets on these screws else the dry rubber will give you leaks again. The valve cover gasket is a rectangular item then there are two gaskets that have the spark plug holes molded in, and the 15 or so grommets.