Reduce Clique to N-Degree-Clique
One almost correct reduction looks like: consider the graph $G'$ that is obtained by adding $N$ new vertices to the input graph $G$. Also, add edges that connect every new vertex to all the vertices in $G$. Its not hard to show that $G$ has a clique of size at least $k$ iff $G'$ has a clique of size at least $k+1$. Note that the degree of each new vertex equals the number of vertices of $G$, which not necessarily at least $N$. But this is a minor issue that I leave for you to fix (along with proving correctness).