As a quick answer, you need to go pretty fast. Like in the range of 1400 to 2400 MPH. That is pretty fast. Supersonic!
The average overall speed would be just over 1600 MHP based on a distance of around 2500 miles in about an hour and a half. At first the speed would be higher, in the 2400 MPH range. That is because the shadow is hitting at an angle. It is more oblong in shape (like a football) but that doesn't matter much because we need to keep up with it. The shadow speed slows as the eclipse reaches the part of the globe that sticks out the most, near Illinois/Kentucky/Tennessee. At that point it is "only" doing about 1400 MPH.
Is there an aircraft that can do it?Turns out yes, there are several aircraft that might have a chance. The problem is how far can they run at that speed. Most might need to slow way down to fuel up the tanks and by then the shadow has out run them.
In 1973 a prototype of the SST Concorde raced the shadow of the Moon across Africa. It was able to keep up for an astounding 74 minutes before breaking off. The thing is, there are no windows looking up from the Concorde, just a small area for instruments to be placed. They could see out the window, not the eclipse, but the shadow sweeping over the land below.
What about those super spy planes, can't they go that fast? Again, they can but not for long. They go even faster otherwise they might burn up the fuel too quickly. So while they can run at the speed for a short while they would either have to slow down and fuel up or jump way ahead and make a loop about the shadow thus not staying under it the whole time.