Where the Control Resonant dog appears
Downtown, Central, the Evacuation Zone, the West Incursion Zone, The Park, the Underpass and the Unknown. In each, find the dragon toy nearby and drop it on her bed. The toy respawns if you carry it too far, and later spots need Shift and Reach to get to.
Finishing the Mysterious Dog side story
After the seventh toy, go to The Gap — the dog is waiting there. You get the Dog’s Best Friend trophy/achievement and the recipe for the Trusted Spanner artifact (+15% Raw Damage for Dylan).