Okay, this might be more depth than you were looking for, but here ya go.
I have the regular smallest size brushes that you can find at a craft store. The one I used is specifically a Connoisseur brand round in size 0. To make really small lines, water your paint down, angle your brush to get more of a point, and hold your brush farther back on the handle (this is why tiny striping brushes with really long bristles are a thing: better control for really tiny marks!). And honestly, getting really tiny lines like her eyelashes can come down to luck. (I was shocked!)
The egg is wooden, so the base color was a medium brown. (Handy for values) The order was sketch, background, hair, skin, details. I did the lighter part of the hair first, and made the rainbow gradient. The darker parts of the hair are basically the color I mixed minus the white, plus water. However, I typically start with the middle tone! (This is a preference, but it’s easier for me to figure out values that way. Some people do a whole black and white under painting before they add color!)
The skin has a lot of color variation since I had a hard time mixing the exact color I wanted. (And I tend to mix directly on the piece)
The darkest value is a warm blue plus a red closer to yellow than purple. This makes a dark, desaturated purple. The lighter purple line on the strand of hair by her ear? Same color as the dark lines, just much more watered down.
Uhh, I’m an art education major, so I’m happy to answer any other art questions in way too much depth...