Heat 1 tablespoon of the oil in a medium saucepan over medium. Add bell pepper; cook, stirring occasionally, until softened, about 6 minutes. Add shallot, lemongrass, and ginger; cook, stirring often, until fragrant, about 1 minute. Add curry paste; cook, stirring constantly, until slightly caramelized, about 30 seconds.
Add chicken broth, coconut cream, and 1/8 teaspoon of the salt; bring to a boil over medium-high, scraping bottom of pan to release browned bits. Reduce heat to medium-low and simmer, uncovered, until slightly reduced, about 5 minutes. Add bok choy; cook, stirring often, until tender, about 4 minutes.
Meanwhile, heat remaining 1 tablespoon oil in a large skillet over medium. Sprinkle chicken breasts with pepper and remaining 1/8 teaspoon salt. Cook until a thermometer inserted in thickest portion registers 165°F, 6 to 7 minutes on each side. Let rest 5 minutes before slicing.
Prepare rice according to package directions. Divide rice, curry sauce, and chicken among 4 shallow bowls. Garnish with basil and lime wedges.