Heat oil in a large pot over medium heat. Add garlic; season with salt and pepper. Cook, stirring occasionally, until garlic is lightly fried and turning golden brown, 2-3 minutes. Using a slotted spoon, remove garlic to a plate lined with a paper towel. Season with salt; set aside.
Add shallots to same pot; season with salt and pepper. Cook, stirring occasionally, until shallots are lightly fried and turning golden brown but are not yet crisp, 3-4 minutes.
Add mushrooms; season with salt and pepper. Cook, stirring occasionally, until beginning to brown and totally softened, about 4 minutes. Add broth; season with salt and pepper. Bring to a simmer; cook, uncovered, until flavors get to know each other and broth tastes a little shalloty and a little mushroomy, seasoning with additional salt and pepper as needed, 30-40 minutes.
Once broth is as good as can be, add sliced celery and chicken; cook until celery is just tender and chicken is warmed through, 3-4 minutes. Season with fish sauce and lots of lime juice; stir in celery leaves, cilantro and generous spoonful of hot sauce. Top with toasted garlic just before serving.