Heat oil over medium heat, in medium nonstick (10-inch) sauté pan. Add onions and potatoes and cook until onions are softened, 8 minutes, stirring occasionally.
Stir in broccoli and cauliflower; cook until vegetables are tender, 10 minutes, stirring occasionally.
Reduce heat to low and pour in egg mixture. Drop small spoonfuls of goat cheese on top. Cook about 2 more minutes on the stove.
Transfer to oven and bake 20 to 25 minutes, or until the top is golden brown and the eggs have just set in the center. Note that if you are using a skillet with a nonmetal handle, wrap it with double-thickness aluminum foil before placing in oven. Remove from oven and slide out of pan onto a cutting board. Cut into wedges and serve warm.