To fully realize robots in household environments, robots would need to be able to plan and execute a variety of tasks autonomously. However, task and motion planning for multi-step manipulation tasks is still an open challenge in robotics, particularly for long-horizon tasks in dynamic environments. This work proposes an integrated task and motion planning (TAMP) robotic framework for real-world cooking tasks using a dual-arm robotic system. Our framework integrates PDDLStream, an existing TAMP framework, with the MoveIt Task Constructor, a multi-stage manipulation planner, to enhance multi-step motion planning for interdependent tasks. We augment our framework with various cooking-related skills, such as object fixturing, force-based tip detection, and slicing using Reinforcement Learning (RL). As a motivating case study, we tackle the long-horizon task of preparing a simple cucumber salad, consisting of slicing and serving it on a plate. We demonstrate our framework both in simulation and real robot demonstration.