For the toast: Preheat the oven to 400 degrees F. Line a baking sheet with parchment.
Put the bread cubes on the prepared baking sheet, drizzle with the oil and season with salt and pepper. Bake until toasted, 10 to 15 minutes. Let cool completely.
For the dressing: In a large bowl, whisk the olive oil, lemon juice, mustard, capers and everything bagel seasoning.
For the salad: Add the tomatoes, cucumbers, avocado and onions to the dressing and toss. Add the bread cubes and basil and toss again. Let stand at room temperature for about 20 minutes before serving. Garnish with the feta and halved eggs.