Put oven rack in middle position and preheat to 375 degrees.
Whisk together first 5 ingredients using ? teaspoon salt until combined.
Whisk in cheddar and 3 Tablespoons of parmesan.
Heat oil in a 10-inch skillet with an oven-proof handle until hot, but not
smoking. Saute shallot, stirring occasionally, until browned, about 3
minutes. Add hash browns and remaining 1/2 teaspoon salt and cook,
stirring occasionally, 3 more minutes.
Spread potatoes evenly in skillet and cook, undisturbed, for 2 minutes.
Remove from heat and spread broccoli over potatoes, then pour in egg
mixture. Sprinkle with remaining parmesan. Bake until set 2 minutes from
edge but still slightly wobbly in center, 12-15 minutes.
Turn on broiler. Broil 4 to 5 inches from heat until top is set, puffed,
and golden brown.
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