Preheat broiler with rack 5 inches from heat source. Place garlic and ginger, cut side down, on a small baking sheet. Broil in preheated oven until charred, 5 to 7 minutes.

Meanwhile, trim scallion ends and cut white parts from green parts. Tie together white parts of scallions and cilantro stems using kitchen twine. Thinly slice green parts of scallions, and reserve for garnish.

Whisk together broth and bouillon base in a large pot until combined. Stir in chicken, rice, scallion-cilantro bundle, and charred garlic and ginger. Bring mixture to a boil over high; reduce heat to low, and simmer, stirring occasionally (to prevent rice from clumping or sticking to the bottom of the pot), until thickened and liquid is evaporated, 45 to 50 minutes.

While congee simmers, stir together oil, turmeric, and chiles (if using) in a small saucepan over medium; bring to a simmer, and cook, swirling occasionally, until turmeric softens and oil is a bright yellow color, about 5 minutes. Remove from heat; let stand until room temperature, about 20 minutes. Pour through a fine mesh strainer into a bowl, discarding solids; set turmeric oil aside.

Transfer chicken to a clean cutting board. When cool enough to handle, shred meat from chicken; stir meat into congee. Discard skin and bones. Remove and discard scallion-cilantro bundle, garlic, and ginger.

Stir coconut milk and white parts of bok choy into congee; bring to a simmer over medium. Cook, stirring occasionally, until bok choy is tender and rice is creamy, about 5 minutes. Add green parts of bok choy; cook, stirring until wilted, about 1 minute.

Garnish congee with sliced scallions, chopped cilantro leaves, and lime wedges; drizzle with reserved turmeric oil. Serve hot.