Do egg dishes need to be refrigerated??

I made an oatmeal casserole, it has eggs and milk. I baked it for 35 minuets and left it out over night... is it safe to eat? Im breastfeeding my 11 day old and I'm worrisome about everything! It is apparently suppose to stay gooey like oatmeal which it was? I ate like four bites or so before thinking about it but I don't refrigerate cakes or anything isn't that the same thing?? Ugh!! I'm stressed!

2 cups gluten-free rolled oats
1/3 cup brown sugar
1 teaspoon baking powder
1 teaspoon cinnamon
1/2 teaspoon salt
1 cup walnut pieces
1 cup raspberries {any berries work}
1/2 cup milk chocolate chips
2 cups milk
1 large egg
3 tablespoons butter, melted
1 tablespoon vanilla extract
1 ripe banana, peeled, 1/2-inch slices

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