Dry-roast the coriander seeds, cumin seeds, and white peppercorns in a pan over medium heat until fragrant, about 2-3 minutes. Grind them finely using a spice grinder or mortar and pestle.
In a food processor or blender, combine the roasted ground spices, shallots, cilantro stems, garlic, kosher salt, sugar, and turmeric. Pulse to form a rough paste.
Pour in the evaporated milk, coconut milk, sweetened condensed milk, and fish sauce. Blend again until you get a smooth, creamy marinade.
Place the chicken thigh pieces in a large bowl or zip-lock bag. Pour the marinade over the chicken, mixing well to coat. Cover or seal and refrigerate for at least 3 hours, ideally overnight.
Heat your grill or grill pan to medium-high. Lightly oil the grill grates to prevent sticking.
Remove the chicken from the marinade and thread pieces onto soaked skewers, packing them evenly but not too tightly.
Place skewers on the grill and cook for 5-7 minutes per side, turning carefully to get an even char until cooked through, about 30-50 minutes total.
While the chicken cooks, prepare the peanut sauce by combining the coconut milk, Thai red curry paste, sugar, tamarind paste, and fish sauce in a small saucepan over medium heat. Stir until thickened and fragrant, about 5-8 minutes. Stir in the ground peanuts.
For the cucumber relish, mix vinegar, water, sugar, and salt until the sugar dissolves. Add cucumber, shallot, and sliced pepper. Toss gently and let sit for at least 15 minutes.
If using, butter the slices of bread and toast in a skillet or oven until golden brown.
Serve the grilled chicken satay skewers on a platter with peanut sauce on the side, adding cucumber relish and toasted bread if using.
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Notes
Store leftovers in an airtight container in the refrigerator for up to 3 days. Reheat the chicken in the oven or a skillet.