Why your dog cannot resist the moving cat
A chase rarely starts with the sprint itself. It starts with a hard stare, a lean forward and a tail that goes stiff and high, and a running cat triggers that sequence in seconds because movement is what a dog's prey drive is built to follow. Your own cat makes it harder, not easier, because she runs the same route through the same hallway every day, so your dog learns exactly where the fun begins. Shouting once he is already running teaches him almost nothing about the cat; he is too aroused to connect your voice with his choice. The workable point is the stare, while he can still think, choose and turn away.
Set up the room so the cat never has to flee
For the first stage, put a baby gate or screen between your dog and the cat and keep your dog on a light house lead you can pick up calmly. Start at the far end of the room, where he notices her but can still eat a treat and respond to his name. The cat needs her own side of that barrier to be genuinely good: a high perch she can reach without crossing the floor, her scratching post, water and a litter tray your dog cannot reach. If she has to dash past him to reach any of those, she will keep running, and every dash rehearses the chase you are trying to end.
Reward the look-away until it becomes his habit
Watch for the glance, not the lunge. The second your dog looks at the cat and then offers you eye contact, or even flicks an ear toward you, mark it and feed a small, high-value treat right by your leg so his body turns with his head. If he fixes, leans and stops taking food, you are too close; add distance rather than repeating his name louder. Keep sessions short and end while he is still succeeding. Over many brief sessions, move a step closer only when the previous distance looks boring to him, and practise in the hallway and garden as well as the living room, because a dog who is calm in one room has not yet learned the rule in another.
When to separate completely and get help
Some pairings need management for much longer, and a few need it for good. If your dog has grabbed, cornered or injured the cat, or he shakes and drools and cannot eat at any distance from her, keep them fully separated and bring in a qualified behaviour professional who can watch the body language in person. A cat who stops eating, hides all day or starts toileting outside her tray is telling you the current arrangement frightens her, and her side of the plan matters as much as his training. Until a professional advises otherwise, doors and gates stay closed between them whenever nobody is actively supervising. As the look-aways become reliable, KinsLe helps by holding the shared practice plan in Care Teams, where each family member can see which distance stage your dog has reached and record completed sessions with a timestamp, so the cat gets the same careful setup from everyone.