Marinate the pork chops in 1/2 cup of hoisin sauce for 1/2 an hour. Preheat a grill. Place the pork chops on the hot grill and cook the pork chops, turning frequently and basting with the remaining hoisin sauce on each side until the pork is cooked through, approximately 15 to 20 minutes.
For Sweet and Sour Lychees: Heat the vegetable oil in a saute pan over medium-high heat. Add the sesame seeds and lightly toast. Add the lychees and the vinegar, cook for 30 seconds. Add the onions, toss and cook for an additional 30 seconds. Add the sesame oil and season with salt.
For the Lychee Cucumber Salad: Cut the cucumber in half lengthwise, deseed with a spoon, quarter and dice 1/4-inch thick. In a bowl, mix the cucumber, halved lychees, cilantro stems, and chile. Add the lime zest, lime juice, sugar, salt and vinegar, and toss to combine. Before serving, garnish the salad with the cilantro leaves.
For the Potato Pancakes: Preheat the oven to 350 degrees F. In a bowl, beat the egg and 1/2 cup milk. Add the flour and mix well. Add the garlic powder, salt, and pepper. Add the other 1/2 cup milk and mix well. Add the cooked mashed potatoes and mix well. Add the green onion and incorporate into the mixture.
Heat the vegetable oil in a frying pan over medium-high heat. Once the pan is hot, spoon the potato mixture into the pan forming 4 pancakes. Add the butter to the pan. Flatten the pancakes for even cooking, cook 4 minutes, then flip. Place the pan in the oven for 5 minutes, or until the potato pancakes are brown and cooked through.
For the Lychee Cocktail: Place the ice in 4 glasses. Divide the soda water equally among the glasses. Repeat with the vodka and lychee syrup, pouring them over the ice and soda water. Divide the cranberry juice equally among the glasses. Place one whole lychee in each glass for garnish.