DIAGNOSIS The brown splatter on the flat, rectangular metal panel on the ceiling is baked-on food hitting the waveguide cover. That panel protects the magnetron waveguide, and it's safe to clean, but you need to be gentle.
FIX The safest method is steam. 1) Put a microwave-safe bowl of water with a tablespoon of vinegar or lemon juice inside. 2) Heat it on high for 5-7 minutes until the window is very steamy and the water is boiling. 3) Let it sit for another 5 minutes with the door closed to loosen the grime. 4) Carefully open the door (mind the steam), remove the bowl, and wipe the ceiling and all interior surfaces with a soft cloth or sponge. The gunk should come off easily. For stubborn spots, use a mild dish soap solution, but avoid abrasive scrubbers.
IF THAT DIDN'T WORK For persistent, carbonized spots, make a paste of baking soda and water. Apply it gently to the spot on the waveguide cover, let it sit for 15 minutes, then wipe. Do not use metal scouring pads or sharp objects, as you can puncture the thin metal cover, which would require immediate replacement for safety.
HEADS UP Never spray cleaners directly onto the interior, especially the waveguide cover or the door seals. Spray the cloth first, then wipe. Liquid can seep into seams and electrical components, causing damage or a foul smell when heating.