Nigel Lythgoe Gives Props to Katy Perry, Throws Shade on Simon Cowell

Nigel Lythgoe clearly has beef with Simon Cowell, because he went after him ... after Simon went after the "American Idol" reboot. We got the longtime EP of the OG "Idol" Friday on his way to E Baldi in Bev Hills, and he thinks the new judge's…