Heat 3 tablespoons of the vegetable oil in a large sauté pan or wide, shallow pot over medium-high heat. When the oil is hot, sauté the onion, jalapeño and celery for 2 minutes, stirring once or twice. Add the garlic and sauté for another minute or two.
While the vegetables are sautéing, mix the tomato paste, broth and vinegar until they are combined. Add to the pan with the vegetables and bring to a boil. Add the rosemary and a pinch of salt.
As the sauce is boiling, slice the okra on the bias to create diagonal pieces. The reason for diagonal slicing is to expose as much of the interior of the okra as possible. Wait to cut the okra until the last minute because it helps make them less slimy.
Heat another pan over high heat for a minute or two. Add the remaining oil and get it almost smoking hot, which should take 1-2 minutes. Add the sliced okra and spread out in a single layer in the pan. Let the okra brown for at least a minute before you move them. The goal is to cook the okra quickly at very high heat without moving it too much. The high heat sears the okra and helps limit the slime factor. Sear the okra for 3-4 minutes, stirring only 2-3 times.
As soon as the okra is done, add it to the boiling sauce. Add the diced tomatoes and reduce the heat to a simmer. Cook for 5 minutes, no longer. The tomatoes should still be a bit firm, and you don’t want to cook the okra to the point it starts releasing slime.
Turn off the heat, grind black pepper over everything and taste once more for salt. Add if needed.
Serve over steamed rice or with lots of crusty bread.