Cut shallots into 1/8-inch-thick slices. In a heavy 10-inch skillet cook shallots in oil over moderate heat, stirring occasionally, until golden, 15 to 20 minutes. With a slotted spoon transfer shallots to paper towels to drain and season with salt. Reserve 3 tablespoons oil for dressing salad and cool shallots to room temperature. Shallots may be fried 2 days ahead and kept in an airtight container at room temperature.
Just before serving, in a large bowl toss together greens, sprouts, reserved oil, vinegar, and sea salt.
