Process saltines in food processor until finely ground, about 30 seconds; transfer to bowl and set aside.
Pulse onion in now-empty food processor until finely chopped, 5 to 7 pulses.
Arrange chicken, skin side down, in single layer in bottom of Dutch oven.
Add broth, 1 teaspoon salt, and onions and bring to boil over medium-high heat.
Reduce heat to low, cover, and simmer for 30 minutes.
Remove pot from heat and transfer chicken to plate to let cool slightly, about 15 minutes; cover pot to keep stew warm.
Discard chicken skin and bones.
Tear chicken into large pieces.
Transfer chicken to clean food processor and pulse until finely chopped, about 7 pulses.
Whisk chopped chicken, saltine crumbs, milk, cream, butter, cayenne, 1 teaspoon pepper, and ½ teaspoon salt into stew.
Heat over medium heat until warmed through but not boiling, 3 to 5 minutes.
Season with salt and pepper to taste.
Serve with extra saltines.
