Preheat the oven to 300°F. Core the tomatoes, then cut them in half horizontally. Put the 8 halves, cut side up, in a baking dish just large enough to hold them. Sprinkle generously with salt. Combine the olive oil, oregano (crumbling the herb between your fingers as you add it), and garlic, then spoon the mixture over the tomatoes, dividing it evenly.
Bake the tomatoes, basting every half hour or so, until they are very soft and just beginning to caramelize, about 3 hours.
Using an offset spatula, transfer the tomatoes to a serving platter, spooning any pan juices over them. Grate the feta over them. Serve warm, not hot.