If we had a better idea of the speed, it'd be easier to make a guess. However, I'm not sure a foreign military aircraft would be "very large", given how risky that would be, I'd envision it as a fighter jet or a Blackbird-type of aircraft.
The ATC commentary suggests a fast relative speed, in reference to other traffic only. So, in the 600MPH to 700MPH region, it would be going faster than most air traffic, but wouldn’t cut the noisy wake of a sonic boom overflight, which means, in my mind it wasn’t going “very fast” or fast enough to put it in the same league as what qualifies for fast for any fighter or other fast military plane.
The “large” description is also vague, but suggesting as big as a passenger jet is my estimate. Perhaps not freakishly large like a KC-10 Extender, C-5 Galaxy or a C-17 Globemaster III [0,1,2]. Large, in aviation, often means “not a 2 passeneger Cessna,” to evoke not scale relative to other humans, but scale of catastrophe if it fell into a neighborhood or if a pilot had to think about colliding with such a thing, which would “win.”
That said, the white paint job sounds uncharacteristic of military/government (especially skunkworks) operations, and the expanding attention of the incident, points to unexpected events.
Another hypothesis is high level leadership asking questions about actual preparedness for something that fits the profile of a sudden terrorist hijacking incident, based on peered civil aviation inputs alone, along with an undisclosed fire drill.