Break up the cold rice with your fingers or a fork so there are no large clumps. Mix kecap manis, soy sauce, and shrimp paste, if using, in a small bowl.
Heat 1 tablespoon oil in a large wok or skillet over medium-high heat. Add the chicken in a single layer and cook, turning occasionally, until browned and an instant-read thermometer inserted into the thickest pieces registers 165°F (74°C), about 5 to 7 minutes. Transfer to a plate.
Add the remaining oil. Scramble the eggs until just set, then transfer them beside the chicken.
Add shallots and chile to the pan and cook for 1 minute. Stir in garlic for 30 seconds. Add rice and spread it across the hot pan; let it sit briefly between tosses so some grains toast.
Return chicken and eggs to the pan. Pour in the sauce and toss until the rice is evenly coated and hot, 2 to 3 minutes. Fold in scallions.
Serve immediately with cucumber and lime wedges.
