Preheat the oven to 180C/350F/Gas 4.
Heat one tablespoon of the oil with the butter in an ovenproof frying pan and fry the sweet potato and garlic for 2-3 minutes, or until beginning to brown. Transfer to the oven to roast for 10-12 minutes, or until the sweet potato chips are golden-brown and crisp.
Heat the remaining oil in another pan and fry the green beans and a third of the shallots for 1-2 minutes. Add the white wine vinegar, then transfer the pan into the oven for 8-10 minutes, or until the beans are tender but still with some bite.
Toss the remaining shallot rings in the flour. Half-fill a deep-based pan with the vegetable oil and heat until a cube of bread dropped in sizzles and turns golden-brown in 30 seconds (CAUTION: hot oil can be dangerous. Do not leave unattended). Fry the shallot rings in small batches for 4-5 minutes, or until golden-brown and crisp. Remove with a slotted spoon and drain on kitchen paper. Season, to taste, with salt and freshly ground black pepper.
To serve, place the sweet potato, garlic and green beans onto a plate and scatter over the cheese. Garnish with the shallot rings.