Place potatoes into a large pot and cover with salted water; bring to a boil. Reduce heat to medium-low and simmer until tender, about 20 minutes. Drain.
Allow potatoes to cool lightly and cut in half.
Preheat the oven to 400 degrees F (200 degrees C). Lightly brush a baking sheet with olive oil.
Combine tomatoes, feta, and ramps in a bowl; season with salt and pepper. Place about 1 tablespoon of feta mixture on top of each potato half and transfer to the prepared baking sheet.
Bake on the center rack in the preheated oven until cheese is lightly browned, about 20 minutes.