Preheat oven to 200ºC.
Toss the quartered shallots with the olive oil, balsamic vinegar and thyme in a small baking dish. Season with salt and pepper. Bake for 40-50 minutes, stirring every 15 minutes or so, until the shallots are tender and beginning to brown. Set aside.
When the shallots are nearly ready, heat a wok on a medium high heat and add the vegetable oil and shredded sprouts.
Stir fry for around 5 minutes, then add the chestnuts, lemon juice and salt and pepper. Stir fry for a further minute, then take off the heat. Stir through the shallots and serve.