I was away for several days, during which we had below zero temperatures. When I returned home I discovered standing water in both sinks in my kitchen. I called a drain guy to come and check for a clog. He took over the pipe to the wall and put the snake in. He could only get it to go about 3 ft before it stopped. It made a grinding noise while he tried to make it go further. He then informed me that the drain pipe was frozen. I find this weird because the hot and cold water work fine.

Could the pipe actually be frozen, or could this be a bad clog?

Get your biggest pot and boil water in it and dump it in your sink. If it is frozen water the boiling water should melt it. If the boiling water seems to work some, when the water is not in the sink anymore boil another pot of water and keep trying it if it seems like that is working.