Slice the shallots thinly, lengthwise.
Set a small fine-mesh strainer on top of a heatproof bowl and place both close to the stove.
With a room temperature small pan, add the oil and the sliced shallots, mix together then heat the pan over medium heat. Stir the shallots around a bit to separate them.
Cook, stirring occasionally, until the oil becomes hot and starts sizzling, about 5 minutes. Turn down the heat to medium-low and continue to cook, stirring occasionally as the shallots around the edges of the pan tend to cook faster than those in the middle.
After 5 minutes, the shallots should be the color of honey (if not, turn the heat down to low and continue to cook for 1 to 2 minutes longer). Immediately remove the skillet from the heat and pour the contents through the prepared strainer.
Let both the crispy shallots and the oil cool completely before storing them in 2 separate airtight containers at room temperature. The crispy shallots keep for 3 weeks and the oil keeps for 2 months.