In a large bowl, combine 2 cups boiling water with the sugar and salt and stir to dissolve. Let cool completely. Add the chicken breasts, cover, and refrigerate for 2 to 3 hours.

In a medium skillet, lightly toast the cumin, coriander, and caraway seeds over medium heat, stirring often, until fragrant, about 2 minutes. Grind the seeds to a powder in an electric spice grinder and transfer to a medium bowl. In the same skillet, toast the chiles over medium heat, pressing on them with a spatula, until fragrant and darker in spots, about 1 minute per side. Remove the stems and seeds and grind in the spice grinder--you'll need 3 tablespoons ground chile. Add the ground chile, cayenne, garlic, and 1 teaspoon salt to the ground spices, and then stir in the oil and vinegar. Taste and adjust the seasoning with more salt or vinegar if necessary. Stir before using.

Prepare a medium gas or charcoal grill fire. While the grill heats, remove the chicken from the brine, pat dry with paper towels, and let sit at room temperature.

Toss the chicken with 2 tablespoons of the chile oil. Grill the chicken without moving until golden brown grill marks form, 4 to 6 minutes. Flip the chicken and grill until just cooked through, 4 to 5 minutes. Transfer to a cutting board and let rest for a few minutes. Holding your knife at an angle, cut the breasts crosswise into 1/2 -inch-thick slices and let cool to room temperature. Serve with the remaining chile oil on the side.

The chile oil can be made up to several hours in advance, or up to a day ahead and stored in the refrigerator. The chicken can be grilled up to 6 hours in advance. Refrigerate and return to room temperature for serving.