Heat the oven to 375°F (190°C) and lightly grease an 8-inch square baking dish. Press softened cream cheese against a large bowl until smooth.
Mix in the yogurt in three additions until fully combined. Stir in the hot sauce, garlic powder, onion powder, dill, celery seed, pepper, and salt until even.
Fold in the shredded chicken until coated, breaking apart any compact clumps. Reserve ¼ cup of each cheese, then fold the remainder into the chicken mixture.
Spread the filling evenly in the prepared baking dish without tight compaction. Scatter the reserved cheddar and mozzarella over the top, leaving a narrow visible border.
Bake uncovered for 18 to 22 minutes, until the edges bubble slowly and the center reaches 165°F (74°C). Avoid vigorous bubbling — remove the dish as soon as the center is hot.
Rest for 5 minutes. Meanwhile, combine the celery, scallions, parsley, lemon juice, and salt. Spoon the celery mixture across the center of the rested dip and serve warm with celery, carrots, pita chips, or sturdy tortilla chips.