In a large pot or Dutch oven, warm oil over medium-high heat until shimmering. Add mustard seeds and cook until they begin to pop and sizzle, about 30 seconds. (Stand back slightly—they can sputter.)

Add onion and celery, season with salt, and cook until translucent, about 5 minutes. Add garlic and ginger and cook until fragrant, about 2 minutes.

Add turmeric, curry powder, red pepper flakes, and cumin. Cook for 1 minute, stirring constantly, until the spices smell toasty, not raw.

Add carrots, potato, and apple. Cook until just starting to soften, about 2 minutes.

Add tomato paste and flour. Cook, stirring constantly to coat all ingredients and prevent lumps, for 2 minutes. The mixture should look paste-like and slightly darkened.

Add red lentils and vegetable broth. Stir well, scraping up any browned bits from the bottom. Bring to a boil, then reduce heat to medium-low.

Cover and simmer for 10 minutes, then uncover and simmer for 15–20 minutes more, until carrots and potato are tender and lentils have broken down completely. The soup should be creamy and thick. Taste and adjust salt if needed.

Remove from heat. Stir in coconut milk and lime juice. Ladle into bowls and garnish with fresh cilantro. Serve with extra red pepper flakes if desired.