Place Cheddar cheese and cream cheese in the freezer for 20 minutes to firm up before slicing.
Meanwhile, combine olive oil, vinegar, parsley, basil, green onions, garlic, pimento, sugar, salt, and pepper in a lidded jar. Cover tightly and shake vigorously. Set marinade aside.
Remove cheeses from the freezer. Cut block of Cheddar cheese in half lengthwise, then crosswise into 1/4-inch slices. Set aside. Cut the block of cream cheese following the same procedure. Arrange alternating Cheddar and cream cheese slices standing on edges in a shallow baking dish. Pour marinade over cheese. Cover and refrigerate at least 8 hours.
Transfer cheese slices to a serving platter in the same alternating pattern. Spoon marinade over the cheese.