Preheat the broiler. Slice off the top 1/4 inch of each tomato, then scoop out and discard some of the core using a melon baller or small spoon. Cut a thin sliver off the bottom of the tomatoes so they will sit flat. Put the tomatoes on a rimmed baking sheet and toss with the olive oil, red pepper flakes and 1/4 teaspoon kosher salt, then stand them up and press a cube of feta about halfway inside each one. Broil until the cheese is browned and slightly soft, about 2 minutes.
Mix the olive paste with 1 tablespoon water and dot all over a serving plate. Arrange the broiled tomatoes on the plate; drizzle with olive oil and sprinkle with sea salt. Scatter the mint on top. Serve warm or at room temperature.