Downtown PHX breakfast
Ridiculously easy question. Matt's big breakfast is at Roosevelt and 1st Street and has a national reputation. Expect a wait. I think it's walkable but it will only take you 5 minutes via light rail. It's one of the great breakfast places anywhere. BYW in that same area are any number of good restaurants: Pasta Bar, Sens, Roosevelt Tavern, Etc. that put hotel food to shame.
Breakfast availability will depend on the day of the week.
The board favorite would probably be Matt's Big Breakfast, a reasonable walk from the Hyatt at 1st St. & McKinley. If it's a weekend day, be prepared for a wait unless you get there early.
Just down the block from Matt's is the Turf, which serves breakfast Saturday, Sunday, & Monday only. It's not at the Matt's level, but it's acceptable without the long wait.
District at the Sheraton Hotel serves a very nice breakfast buffet with an emphasis on local ingredients. Yes, I know it seems silly to walk from one hotel to another, but I'd rate District as far better than the Hyatt restaurants. In fact, District would be my personal first choice on this list.
Local Breeze serves a nice breakfast. It's a longer walk from the Hyatt (maybe 3/4 mile) than some might be comfortable with, but I'd be willing to do it in the milder morning temperatures.
If it's a weekday, places like Tom's Tavern and First Watch, both on Washington Street, are an option. However, I think both are closed on weekends.