Prep and cook instructions are identical for both chicken options.
Pat the chicken dry with a paper towel. Cut the chicken crosswise into ½-inch-thick strips; season generously with salt and pepper. In a large sauce pot over medium heat, warm 1 to 2 tablespoons oil until hot but not smoking. Add the chicken and cook, stirring occasionally, until browned but not yet cooked through, 2 to 4 minutes. Transfer to a plate. Do not clean the pot.
While the chicken browns, start preparing the vegetables.

• Peel and thinly slice enough onion to measure 1 cup [2 cups].
• Cut the mushrooms into quarters.
• Remove the stems and seeds from the sweet mini peppers; cut the peppers into 1-inch pieces.
• Trim the ends from the Brussels sprouts. Cut the sprouts in half lengthwise; cut any large sprouts into quarters.

In the same pot used for the chicken, if dry, add 2 to 3 teaspoons oil. Warm over medium-high heat until hot but not smoking. Stir in the onion, mushrooms, and lemongrass paste, season with salt and pepper, and cook until the onion and mushrooms start to soften, 1 to 2 minutes. Add 1 cup [2 cups] water and bring to a boil. Reduce to a simmer and cook, stirring occasionally, until the liquid is thickened slightly, 3 to 5 minutes.
Add the sweet mini peppers, Brussels sprouts, and chicken and any accumulated juices and bring to a boil. Reduce to a vigorous simmer, cover, and cook until the vegetables are just tender and the chicken is cooked through, 3 to 5 minutes.
While the tom kha gai simmers, prepare the remaining ingredients.

• Juice half the lime; cut half into wedges for garnish. [Juice 1 lime; cut 1 lime into wedges.]
• Coarsely chop the cilantro.

Remove the pot from the heat and stir in the coconut milk and 1 tablespoon [2 TBL] lime juice; season to taste with salt and pepper.

Transfer the chicken tom kha gai to individual bowls. Garnish with the cilantro and as much sambal oelek as you like. Serve with the lime wedges.