• Pat the chicken dry with a paper towel. Cut the chicken crosswise into 2-inch-thick strips, then cut the strips crosswise into 2-inch pieces. Season generously with salt and pepper.

In a large frying pan over medium-high heat, warm 1 to 2 tablespoons oil until hot but not smoking. Working in batches if needed, add the chicken and cook, stirring occasionally, until lightly browned but not yet cooked through, 2 to 4 minutes. Transfer to a plate. Add more oil between batches if needed. Do not clean the pan.
While the chicken browns, prepare the vegetables.

• Peel and thinly slice enough onion to measure 1½ cups [3 cups].
• Scrub or peel the carrots and trim the ends; cut the carrots on the diagonal into ¼-inch-thick slices.
• Trim the ends from the squash; cut the squash in half lengthwise, then crosswise into ½-inch-thick half-moons.
• Cut the mushrooms into quarters.

In the same pan used for the chicken, if dry, add 2 to 3 teaspoons oil. Warm over medium-high heat until hot but not smoking. Add the onion, carrots, squash, and curry powder, season with salt and pepper, and cook, stirring occasionally, until fragrant and the vegetables start to soften, 3 to 4 minutes.
Add the coconut milk, tomato base, and mushrooms and bring to a boil. Reduce to a simmer and add the chicken and any accumulated juices. Cover and cook until the chicken is cooked through and the vegetables are tender, 3 to 5 minutes. Remove from the heat and season to taste with salt and pepper.
When the curry is almost done, prepare the garnish.

• Strip the basil leaves from the stems; coarsely chop the leaves.

Transfer the chicken curry to individual bowls, garnish with the basil, and serve.