This research proposes the definition of m-level Pizza Graphs which are m-regular graphs. We proved that all m-level Pizza Graphs are 1-edge fault tolerant Hamiltonian graphs. When m is even, we further prove that m-level Pizza Graphs are m-vertex fault tolerant Hamiltonian graphs. We also introduce the concept of pseudo k-vertex fault tolerant Hamiltonicity, and proved that if a m-level Pizza Graph is m-vertex fault tolerant Hamiltonian then it is also pseudo (m+1)-vertex fault tolerant Hamiltonian. For m-level Pizza Graphs with an even m, we obtain bounds for their cycle decomposition and their edge fault tolerant Hamiltonicity. For m level Pizza Graphs with an odd m, we also study their vertex fault tolerant Hamiltonicity the number of disjoint cycles and their edge fault tolerant Hamiltonicity. Finally we suggest the concept of k-layer m-level Pizza Graphs, and that they may be (m+1)-vertex fault tolerant Hamiltonian when k = 2, and (m+2)-vertex fault tolerant Hamiltonian when k is greater than 2.