If you say, “Hey Siri, open Notes,” and have a hand free to tap the Compose icon, then tap the Mic button beside the space bar in the Compose keyboard. Immediately Siri will transcribe indefinitely, until you reach over and tap Done. Unless the screen were to shut off, this is a 3-touch solution.
By means of the Shortcuts app that Apple has released more recently than the other answers on this question, you can make a simple chain from “Dictate Text” into “Create Note”, but you still only have the options to stop it with a pause, a short pause, or a tap, not with a codeword, so for your purposes it’s still a 1-touch solution, not safely handsfree.
Edit: iOS 13 broke this so badly it often stops recording after less than a second's pause between words. However a new answer is possible through the new Voice Control utility iOS 13 has introduced under the Settings > Accessibility menu (scroll down into Physical And Motor section). After downloading and activating it, you can now speak to the Home screen, "Open Notes," then, "Click New Note," and now there are a number of commands it can recognize as ending the continuous dictation by doing things with the text. Details at link:
"When the cursor is in a document, email message, text message, or other text field, you can dictate continuously."