In a medium skillet, heat 1 Tbs. each of the olive oil and butter over medium heat. When the foam subsides, add the onions and cook, stirring occasionally, until very soft and golden, 20 to 25 minutes; reduce the heat if they brown too quickly. Season with salt to taste and set aside.
Meanwhile, in a large saucepan, bring 2 quarts salted water to a boil. Add the parsnips (cut them in half if they don’t fit in the pan) and cook for 3 minutes. Drain, run under cold water to cool them quickly, and drain again very well. Grate the parsnips in a food processor fitted with a medium grating disk. In a medium bowl, combine the parsnips, leek, and egg. Stir in the flour, 1 tsp. salt, and 1/8 tsp. pepper.
In a 10-to 12-inch heavy skillet, heat the remaining 2 Tbs. oil and 1 Tbs. butter over medium-high heat until the foam subsides. Shape the parsnip mixture into four equal balls. Put them in the skillet and press on each with a flat spatula to make a cake about 3-1/2 inches wide. Reduce the heat to medium and cook until browned on one side, 4 to 6 minutes. Turn the cakes over and brown the other side, 4 to 5 minutes. Flip to recrisp the first side, about 30 seconds. Drain briefly on paper towels and then serve while hot, garnished with a large dollop of sour cream and the caramelized onions.