Preheat oven to 425°.
In a 12-in. cast-iron or other ovenproof skillet, melt butter over medium-high heat.
Add shallot; cook and stir until tender, 1-2 minutes.
In a large bowl, whisk eggs and milk until blended.
Whisk in flour, salt and pepper until smooth; stir in thyme.
Pour batter into hot pan; sprinkle with Parmesan.
Bake until top is puffed and sides are golden brown and crisp, 20-25 minutes.
Serve immediately.
