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In a medium pot, combine 1 cup water and the rice wine vinegar, and bring boil over medium-high heat. Stir in the sugar and pickled mango puree, then add the sliced cucumbers. Cook for about 15 seconds and then strain the cucumbers. Set aside to cool.
Heat two nonstick pans over medium-high heat until very hot. Roll the hamachi around in the marinade, making sure it is well coated in oil; reserve the marinade for later use. Place two pieces of hamachi in each pan, presentation-side down, and add a little vegetable oil to the pan around the fish. Cook for 15 to 20 seconds, until a nice crust has formed. Flip the fish and cook for 10 to 15 seconds on the reverse side; remove to a cutting board to rest.
With a very sharp knife, slice each fillet into thin slices. Fan the slices out on four plates. Arrange the pickled cucumbers in ring molds in a cylindrical tower at the top of each plate. Remove the ring molds and top the cucumbers with a few strands of mint chiffonade. Spoon the remaining marinade over and around the fish and serve immediately.