AC's Northern Tier route passes through Odell, IL, about 100 miles from Chicago. You could hop on their, take it to outside Anacortes, WA and then take their Pacific Coast route to Seattle. But it would be a pretty tall order to do it in 35 days. From Muscatine, IA to Anacortes alone is over 2,700 miles. Odell to Muscatine is probably another 200 miles. And it's another solid 2 days to Seattle from Anacortes. Even if you could save a day or two buy skipping the trip into Canada from Cut Bank, MT, you would still need to average close to 90 miles.day with no rest days to make it in 35 days. You could trim a few days off of that by taking the shorter alternative through MN--going from Dalbo, MN to Fargo, ND without going up north to Grand Rapids, MN, but then you would miss Lake Itasca, where the Mississippi River starts. (If you look at the Norther Tier maps detail on the AC web site this will make more sense.)
To give you an idea, when I did Seattle to Maine using the Pacific Coast and Northern Tier routes, I started around May 27th. Got to Fargo, ND on July 6th. Taking the long way through MN took another 2 weeks. Averaged about 60-65 miles/day with a day off every 6th or 7th day.
If it turns out you will be pressed for time, consider taking the train to Minneapolis and starting from there. (Seattle is also convenient for taking the train back to Chicago.)
As for scenery, it all depends on what you consider scenic. I happen to love the rolling farms of ND. Others don't get excited about that. The Northern Tier route uses Going to the Sun Road in Glacier National Park. You can't get much more scenic than that from a mountain perspective. The ride along Lake Koocanusa between Eureka and Libby, MT and the North Cascades Highway in WA are also quite pretty, just to name a few.