I have two solutions for you!
Unfortunately, the one that keeps the aesthetic of a wood frame is not only more complicated, but it also has a large delay between the two doors opening and closing.
This was pretty fun to make, small challenges like these are pretty gnarly
Edit: The noteblock in the second design is NEEDED.
First design regular piston under the lever and a sticky piston pushing the redstone block.
Second design just has a stick piston.