Heat a charcoal grill to medium (about 350°) or set up a camp stove and use medium heat. Warm a large heavy frying pan until hot, then oil bottom and sides. Toast pizza crust (cheesy side down, if there is one), pressing down on edges, until crunchy and golden on bottom, 4 to 5 minutes. Transfer to a cutting board.
Stoke fire with 12 to 15 more briquets if using charcoal. Add 2 tbsp. oil to pan, then onions, sausage, salt, pepper, and thyme. Cook, stirring often, until onions are soft and medium golden brown, 8 to 12 minutes. Remove pan from heat. Scoop onion mixture into a bowl and wipe out pan.
Brush pan with remaining oil. Fit pizza crust into pan with toasted side up. Spoon on pizza sauce and two-thirds of onion mixture, followed by cheeses, remaining onion mixture, and oregano. Return pan to heat. Cook, covered with lid or foil, until cheese begins to melt (check underside to be sure it doesn't burn), 3 to 5 minutes.
Transfer pizza to board. Tent with foil to melt cheese completely, then slice.