For fresh peppers: Light the grill to medium. Place the red peppers on the grill, turning, for about 10 minutes total, until charred on all sides. Remove, place in a shallow bowl, and cover with plastic wrap. You can also do this under the broiler. When the peppers are cool enough to handle, peel and seed in the bowl. Strain the juices and set aside. Cut the peppers lengthwise into strips about 1/8-inch thick. Toss with one tablespoon of olive oil, a little salt, and one tablespoon of oregano.
If using brined Florina peppers, remove 3 peppers from the jar, rinse well under cold water, drain and pat dry. Proceed with the recipe, following the next step, below.
Combine the feta, ¼ cup olive oil, hot sauce, pepper juices (if using freshly roasted peppers), black pepper and lemon juice in the bowl of a food processor and pulse on and off until smooth and spreadable. Set aside.
Brush the bread slices on both sides with the remaining olive oil. Grill for about 2 minutes per side. Remove.
Spread each piece of grilled bread with about one heaping tablespoon of the whipped spicy feta. Top with a few strips of roasted red pepper and sprinkle with additional oregano. Garnish with whole basil leaves. Drizzle with a little extra-virgin olive oil and serve.