Place the shallots, oil, broth, thyme, pepper and salt in a small casserole dish. Cover and roast until the shallots are very soft and brown, about 45 minutes. Remove the casserole from the oven.
Cook the potatoes in a large pot of boiling water until soft, about 20 minutes. Drain and place the potatoes back in the pot over low heat to dry.
Heat the milk over medium low heat. Add it to the potatoes and beat with a mixer. Add the roasted shallots and beat again until potatoes are smooth.