Scrub limes well. Use a peeler to remove only the zest. Set zest aside and juice the limes. Set juice aside.
Heat peanut oil in a heavy skillet until shimmering. Add garlic, onion, ginger and lemon grass; cook, stirring, until tender but not beginning to brown. Add goat meat and reduce heat to medium; cook until all sides are brown but the interior is not cooked. Add lime zests, sweet potatoes, chayotes, coconut, rum and stock/water. Simmer 45 minutes. Add water as needed but take care towards the end of the cooking time, as the final result should be stew rather than soup. You may also use a little more rum, but take care to taste first.
Meanwhile, dice half the red and yellow pepper strips.
At the end of the simmering time, stir in the pepper dice and some of the curry paste. Taste and add more curry paste accordingly. Simmer 5 minutes, stirring occasionally. Stir in coconut liquid; simmer, stirring, for 1-2 minutes. Stir in reserved lime juice and remove lime zest. Turn off heat and taste before adding salt as desired.
Place cooked rice on a platter and make a well in the middle, mounding it around the edges. Put the goat curry mixture into the well. To garnish the top of the stew, alternately arrange strips of red and yellow peppers in spoke formation. Arrange the egg quarters around the edges and arrange or scatter cilantro prettily. Place lime wedges around the edges of the rice.
Diners can squeeze lime juice over their own portions to taste.