In your slow cooker, stir together ½ cup of kimchi, garlic, ginger, oil, chili paste (if using), and soy sauce. Add sirloin and turn to coat.
Cover slow cooker and cook on Low heat for 8 hours, or High heat for 3 hours.
Using a heavy spoon, stir beef, coaxing it into shreds.
Stir brown rice into your slow cooker. Re-cover slow cooker and cook on High heat for 10 minutes more, or until rice is soft.
Refrigerate individual portions of rice and beef, the remaining 1 cup of kimchi, the cucumber slices, sesame seeds (if using), and cilantro (if using), all in separate airtight containers for up to 4 days, or freeze for up to 4 months. Keep lettuce or tortillas separate until ready to serve.
To serve, divide rice and beef among lettuce or tortillas and top with remaining kimchi, cucumber slices, sesame seeds (if using), and cilantro (if using). Roll up the wrap like a burrito.