Make sate sauce. Blend all sate ingredients in a blender until smooth. The sauce can also be used as a dip for chicken sate. Omit this step if using store-bought sate sauce. This can be made up to 2 days in advance. Bring to room temperature before using.
Boil a large pot of water over high heat. Salt generously with kosher salt. Add linguine and cook according to instructions.
Heat oil in a large skillet over medium high heat. Season chicken pieces with salt and pepper. Make sure chicken is dry and when added to the pan, do not move chicken around until you are ready to flip them. When oil is hot, add chicken and cook until browned on one side (about 3 minutes). Flip and add a cheese slice on each breast and continue to cook until cooked through (about 3 more minutes). Set aside.
When noodles are cooked to al dente, drain them and mix them gently with sate sauce in a large bowl until well incorporated.
Serve chicken on top of noodles and garnish with purple basil leaves.