How to Make It
In a medium saucepan, combine the bananas with the vegetable stock, onion, water, pickled jalapeño, garlic and ginger and bring to a boil. Simmer over moderate heat, stirring, until the bananas and onion are very tender, 15 minutes. Transfer to a blender and let cool slightly. Add the ketchup, basil and the jalapeño pickling liquid and puree until smooth. Scrape the banana ketchup into a bowl and season with salt and pepper. Cover and refrigerate until chilled, about 30 minutes.
Bring a large saucepan of salted water to a boil. Peel the yucca and cut into 3-inch lengths. Cut the lengths into 1/2-inch-thick wedges. Cook the yucca in the boiling water until tender, 10 to 15 minutes. Drain well and spread on a baking sheet to cool and dry; discard any stringy pieces of yucca.
In a large saucepan, heat 1 1/2 inches of oil to 360°. Set a rack over a baking sheet. Working in batches, fry the yucca, stirring gently to separate the wedges, until golden brown and crisp, 5 to 7 minutes. Using a slotted spoon, transfer the wedges to the rack and season generously with salt. Serve right away with the banana ketchup and lime wedges.