That's almost exactly what I do except that I keep a small tub of lard to coat the pan after it dries. I like the coat that lard gives the pan.

Here's a little 6” #3 I often use for eggs. I think it's a Lodge. It was my grandmother's.
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Egg and cheese burrito is a staple breakfast for me. Most of the time there's not much to scrub but the water and brush do a fine job.