In a bowl, use a fork or mixer to blend potatoes, chèvre, 1/4 cup cotija, canned chilies, onion, and salt to taste.
Wearing rubber gloves, cut poblanos in half lengthwise. With a knife, cut seed lump from beneath the stem inside each chili, leaving stem end in place (to form a cup). Pull out and discard veins. Fill chilies equally with potato mixture. Set chilies in an oiled 9- by 13-inch casserole, filling it. Cover tightly with foil.
Bake in a 350° oven for 30 minutes. Uncover, sprinkle with remaining 1/4 cup cotija, and return to oven. Bake until stuffing is dotted with brown, 20 to 25 minutes longer.
Serve chilies with a wide spatula. Add salsa and salt to taste.