A commentor asked for advice on this, and I'm happy to give it since there is a SECRET to avoid this dish becoming a big gloppy mess. It involves removing the water. Here's what you do.
I forgot to take a picture of the secret grating technique, so we'll have to make do with a little sketch. The idea is that you grate only the part of the zucchini from the outside, turning as you go. The core containing the seeds that is left gets chucked.
Got that? Good. Now put the shredded zucchini in a colander set in the sink. Toss with a liberal amount of salt (the salt leaches out the water). Let sit for at least 20 minutes.
Place in a clean kitchen towel and squeeze out as much water as you can.
Add a little olive oil in a nonstick or cast iron skillet and heat. Break apart the clumps of zucchini and spread evenly in pan. Cook over a medium flame for several minutes until the bottom of the zucchini is brown. Use a flat spatula to flip zucchini. This is just the way you would cook hashed brown potatoes (stay tuned—I have secrets for making these too).
I like to finish these with a squeeze of fresh lemon.
I hope conservative readers do not mind the inclusion of a liberal amount of salt. Sometimes liberal can be good, you know. As can conservative. Conservative clothes, for example, on my daughter are a good thing. And I like to use a conservative amount of oil when I cook. But I'm a liberal voter, which is why I enjoyed this article from the Rolling Stone.