Place the beets into a large bowl, and drizzle with 1/4 cup olive oil, salt, and black pepper. Lay out 2 large squares of aluminum foil on a work surface, and place 3 beets onto the center of each sheet. Fold the aluminum foil into 2 envelopes, sealing the beets into the packets; place the packets into a baking dish.
Bake in the preheated oven until tender, 1 to 1 1/2 hours. Check for tenderness after 1 hour by piercing a beet with a fork. Open the foil, and allow the beets to cool until they can be handled; peel, and slice.
Lay out the spinach leaves on an attractive oblong-shaped serving platter. Sprinkle pieces of tomato and avocado over the spinach leaves, and top with chopped red onion. Lay the sliced warm beets over the salad, and top with crumbled feta cheese.
Whisk together balsamic vinegar, 1/2 cup of olive oil, and Dijon mustard until smooth; pour over the salad to serve.