Mix the chili sesame sauce together and add 1/3 cup to a bowl with the cucumbers and radishes. Refrigerate until ready to serve. Reserve the remaining 2/3 cup sauce for serving.
Mix all the sesame ginger sauce ingredients together in a bowl and set aside. (You can also add whole cloves of garlic and peeled ginger to a food processor with the remaining ingredients if you prefer.)
Heat a large skillet over medium-high heat before adding the ground chicken. Cook for 3-4 minutes until golden brown on one side before breaking apart to fully cook.
Once the chicken is fully cooked and broken into small crumbles, add the sauce. Cook for an additional 3-4 minutes until the sauce thickens before turning off the heat.
Serve the chicken over rice with the cucumber crunch, chili sesame sauce, sliced scallions, toasted sesame seeds or sesame garlic crunch, and lime wedges.