Preheat the oven to 400°F (200°C). Line a baking sheet with parchment paper and set aside.
In a large bowl, combine the Greek yogurt and olive oil. Add the self-rising flour in two additions and stir with a rubber spatula until the mixture pulls together into a shaggy mass.
Turn the dough onto a lightly floured surface and knead briefly (about 1–2 minutes) until smooth. If the dough feels very sticky, sprinkle up to 1 more tablespoon flour while kneading — aim for a tacky but manageable dough.
Let the dough rest for 8–10 minutes at room temperature; this relaxes the gluten and makes shaping easier.
Divide the dough into 4 equal pieces. Roll each piece into a 6–7 inch rope, then bring the ends together and press to seal, forming a bagel shape. Gently press the center with your thumb to widen the hole slightly.
Arrange the shaped bagels on the prepared sheet about 1 inch apart. Lightly brush each top with the egg white to help the seasoning adhere and promote a glossy finish.
Generously sprinkle Everything Bagel Seasoning over the brushed tops. For extra crunch, press the seasoning gently so it sticks.
Bake for 18–22 minutes, rotating the pan halfway through, until the bagels are deep golden on top and sound hollow when tapped on the bottom.
Remove from the oven and let cool on the sheet for 5 minutes, then transfer to a wire rack. Slice and toast as desired before serving.