Melt coconut oil in large skillet over medium-high heat.
Pat chicken breasts dry. Season both sides with salt.
Place them smooth side down in hot skillet. Sear 2 minutes per side. Add the water. Bring to boil, then reduce heat to simmer. Cover and cook 15 minutes. Note: This won’t seem like enough water, but it is. The chicken will release more liquid as it cooks.
While chicken is cooking, prepare other ingredients.
When chicken is done, remove breasts to cutting board and chop into bite-sized cubes. Reserve pan juices – you’ll be using them later in the recipe.
Melt remaining coconut oil in large pot over medium heat. Add onion and sauté 5 minutes. Add garlic and sauté 30 seconds (just until fragrant). Add sweet potato and asparagus stalks (not the tips), along with bone broth, 1 teaspoon salt, and pan juices saved from cooking the chicken. Add thyme if using. Bring to boil then reduce heat. Simmer covered 15 minutes.
Remove and discard thyme sprigs. Transfer soup to high-speed blender. Add baby arugula and blend until smooth. (You might need to do this in 2 batches).
Return puréed soup to pot. Add asparagus tips. Bring to boil then reduce heat. Simmer 3 minutes. Add chicken to pot and simmer another minute, just until chicken is warmed through.
Stir in lemon juice. Taste, and add more salt if needed.