Pat the duck breasts dry with paper towels. Score the skin in a crosshatch pattern, being careful not to cut into the meat. Season generously with salt and pepper.
Heat the vegetable oil in a large skillet or wok over medium-high heat. Place the duck breasts skin-side down and cook for 6-8 minutes, or until the skin is golden brown and crispy. Flip and cook for another 3-4 minutes for medium-rare, or longer to your desired doneness.
Remove the duck from the skillet and let it rest on a cutting board for 5-10 minutes before slicing.
While the duck is resting, carefully pour off most of the rendered duck fat from the skillet, leaving about 1 tablespoon. Return the skillet to medium heat.
Add the red curry paste to the skillet and cook for 1-2 minutes until fragrant, stirring constantly.
Pour in the coconut milk and chicken broth. Stir to combine and bring to a simmer. Add the fish sauce and brown sugar. Stir until the sugar is dissolved.
Add the snow peas, sliced red and yellow bell peppers, and bamboo shoots to the curry. Cook for 3-5 minutes, or until the vegetables are tender-crisp.
Slice the rested duck breast thinly against the grain.
Serve the Thai red curry duck slices over the curry sauce and vegetables. Garnish with fresh basil leaves and lime wedges.
Notes
This curry is delicious served with steamed jasmine rice.