What would make a snow blower run for 15 minutes then shutoff, restart for 30 seconds and then shutoff again?

run fault codes and look for EVAP system not ventilating the tank properly , causing a drop in available fuel pressure by the fuel pump running in a vacuum

Loosen Gas cap and see if that helps. Many times the Gas tank Vent gets clogged. When engine runs, it creates a suction and starves the engine of gas.

OR
If the engine was stored with gas in it, sometimes the Rubber seat in the carburetor deforms and wont let the gas flow into the Carb Bowl at a fast enough rate to keep up with the running engine. The bowl empties (after 9 minutes) and the engine quits. Takes a while for the bowl to refill (5 minutes).

Ethanol gas is a carburetor kil ler. Your needle valve ( which acts like a turn off switch) may be stuck allowing too much gas to pour from the gas tank into the carburetor.

Usually, if you give it a little gas and hold it there for a bit, it should stay running. Just don't flood it out by keeping the gas on too long. Also... make sure nothing has clogged with snow. It also helps to go easy when you first start your blower. Kind if like you may have to do with your lawn mower. Don't try to tackle the biggest pile of snow too fast. Save that part for after the blower warms up more. Then blow away!!

try running a can of automotive carb cleaner thru the tank of gas the type that says cleans as you drive on bottle this helps sometimes and keeps you from having to remove and tottlay clean out carb from stale gas being left inside it

reminder: allways run seasonal engines till there out of fuel before putting away for season , that way they tend to start right back up in season with a fresh tank of gas !!!

drain gas from carb and tank get some fresh non ethynol gas
while carb bowl is off spray/ clean with carb cleaner spray.
when you drain gas put it in clear container so you can see if there was water or dirt in it. helps to figure out the problem

put in injector cleaner in gas

does it have spark

fuel pump, clogged fuel injectors, bad gas.... could also be a timing problem among other things depending on what you mean by sputtering.... my advice find a shop that will diagnose the car for free( like AAMCO). hope this helps :)