In a bowl, whisk together eggs & milk. Add flour & whisk until incorporated then whisk in parmesan cheese, scallions, parsley, thyme, salt & pepper. Set aside in refrigerator until sauce & filling are made.
Gouda Sauce
In a small saucepan, melt butter; sprinkle with flour & seasonings. Mix well; add milk & broth, stirring until sauce becomes thickened. Blend in cheese; set aside
Shrimp Filling / Baking
Peel & devein shrimp (you can chop into pieces if you prefer). Prepare filling veggies for cooking.
Preheat oven to 425 F.
Place 2 Tbsp butter in each of TWO 7-inch pyrex baking bowls (alternately you can use one 10-inch cast iron skillet). Place bowls in hot oven to melt butter (and heat the bowls for baking pancakes in). Once the butter is melted & the bowls are hot, divide the batter between them. Bake for 25 minutes.