In the Instant Pot (or a large soup pot on the stove), combine the chicken broth, coconut milk, soy sauce, fish sauce, honey, peanut butter, and Thai red curry paste; whisk until smooth.
Add the chicken breasts, sliced cremini mushrooms, chopped red peppers, grated ginger, and minced garlic to the pot; secure the lid if using an Instant Pot.
If using an Instant Pot, cook on high pressure for 10 minutes then release the steam (natural or quick release). If using the stove, bring to a simmer, reduce heat to medium-low, and simmer about 15 minutes until the chicken is cooked through and shreds easily.
Set the pot to sauté (or keep the stove over medium heat). Remove the chicken and shred it with forks or tongs; return the shredded chicken to the broth.
Stir in the ramen noodles, lime juice, baby spinach, and chopped basil or cilantro; let sit 5 minutes or until the noodles are tender and the spinach is wilted.
Ladle the soup into bowls and top with chopped peanuts and a drizzle of toasted sesame oil before serving.