In a small bowl whisk the coconut milk, fish sauce, brown sugar, and 1 tablespoon curry paste until thoroughly mixed. If you would like more heat and flavor, add additional curry paste. Place the chicken strips in a medium bowl and cover with half of the coconut milk mixture (about 1 cup). Toss to coat, and then cover the bowl tightly with plastic wrap. Marinate, refrigerated, for 1 hour to overnight.
Soak the bamboo skewers in water for at least 30 minutes.
Place the remaining coconut milk mixture in a small saucepan with the peanut butter and whisk to combine. Bring to a simmer over medium heat, and then reduce the heat to low and gently simmer for 3 minutes, whisking often. Cool, and then pour into a serving bowl.
In a medium bowl whisk the vinegar, sugar, and salt until the sugar dissolves. Add the cucumber, shallot, and jalapeño and toss well to coat. Let the salad sit at room temperature for at least 15 minutes before serving.
Pat the excess marinade off the chicken strips with a paper towel. Brush the chicken strips on both sides with oil, and then skewer the strips lengthwise, threading in and out on the skewers.
Prepare the grill for direct cooking over medium heat (350° to 450°F).
Grill the chicken over direct medium heat, with the lid closed, until fully cooked in the center, 8 to 12 minutes, turning once. Remove from the grill, top the salad with the peanuts and cilantro, and serve immediately with the peanut sauce.