Have you tried just writing out the whole command in a single string?
{
"cmd" : ["gcc $file_name -o ${file_base_name} && ./${file_base_name}"],
"selector" : "source.c",
"shell": true,
"working_dir" : "$file_path"
}
I believe (semi-speculation here), that ST3 takes the first argument as the “program” and passes the other strings in as “arguments”. https://docs.python.org/2/library/subprocess.html#subprocess.Popen