One-check is a mate in 5 moves for White, starts with 1.e3. Example 1...e6 2. Qh5 g6 3. Qe5 (threat Qxe6+) and now 3... Be7 4. Qg7, or 3... Qe7 4.Qxc7, or 3...Ne7 4.Qc5
Competitive One-Check Chess

I saw that line too. I analyzed this on Lichess for a long time, and there is actually a mate in 4 for White with 1. Nc3.
Also 1. e3 Nf6 stops your idea of Qh5, and the suggested follow-up is 2. Na3

There is no need for long time analysis, have compiled an engine Onecheck.exe based on Fairy-Stockfish and it finds the mate in 5 in less than 1 second (you can download it at
https://filehorst.de/d/dzejGbfk
it needs a GUI like Winboard where you chose the variant 1check )
Your move 1.Nc3 starts also a mate-in-5 (not mate in 4), and 1.e3 Nf6 2.Na3 starts also a mate in 5

Awesome, thanks for the link! I have been doing stuff by hand and its pretty fun but also using a computer is cool to check for mistakes. I'll take a look.
Anyhow, your engine might have a problem if it says Nc3 is mate in five. How does Black stop mate in four?
After playing three-check chess, I became very interested in analysis of one-check chess. Here is my idea for competitive one-check chess:
Since both players play each side, it should be fair (at casual levels), although the player who is White second can get some experience from the first game.
I also think coming up with one-check positions itself would be fun to do.
One-check chess is supposed to be a forced win for White. (I haven't found the forced win yet, please comment if you know!)