Set oven rack about 6 inches from the heat source and preheat the oven's broiler. Line a baking sheet with aluminum foil; set a baking rack on top.
Rub salt and pepper over the pork ribs; place on the prepared baking sheet.
Cook in the preheated oven, flipping once, until meat pulls away easily from the bone, 7 to 10 minutes per side. An instant-read thermometer inserted into the center should read 145 degrees F (63 degrees C).
Heat oil in a large skillet or wok over medium heat. Add celery, ginger, and garlic; cook and stir until fragrant, 2 to 3 minutes. Reduce heat to low. Mix in soy sauce, chile pepper, sake, tomato puree, sugar, balsamic vinegar, and salt; stir until sauce is combined. Remove from heat.
Black vinegar can be substituted for the balsamic vinegar if desired.
You can bake the ribs on a baking sheet lined with aluminum foil, for 20 minutes, flipping halfway, at 350 degrees F (175 degrees C). An instant-read thermometer inserted into the center should read 145 degrees F (63 degrees C).