In a large jug, whisk the crumbled stock cube, the tomato purée and tatlı biber salçası into the boiling water until dissolved. Leave to one side.
Add 3 tablespoons of the olive oil to a large pot and place on the stovetop over low heat.
Once hot, add the vermicelli to the pan and fry for a couple of minutes, stirring continuously so that each strand browns evenly.
Once the vermicelli strands are a rich golden-brown color, take the pot off the heat. The vermicelli will continue to darken a little once removed from the heat.
Allow the vermicelli to cool down in the pot for a few minutes (to prevent the oil and liquid from reacting and dangerously splattering when the stock is added), then carefully pour in the stock mixture.
Place the pot back on the stovetop over medium heat, stir through the dried mint, pul biber, sea salt flakes and coarse black pepper, bring the soup up to a boil, then simmer on low-medium heat for 10-12 minutes until the vermicelli is cooked but still has a little bit of bite to it (add another 1/2-3/4 cup, 120-180 milliliters of boiling water to the pot while it's cooking if you feel like the soup needs loosening a little).
Brush a small frying pan with the remaining 1 tablespoon of olive oil and place the pan on the stovetop over medium heat.
Pat the Hellim cubes dry with paper towels and carefully add the cubes to the hot oil.
Cook the cubes for a couple of minutes, ensuring to turn them frequently until the cubes are a lovely golden-brown all over. Remove to a plate lined with paper towels (to soak up excess oil).
Once the soup is ready, stir through the lemon juice, take the pot off the heat and let it rest for 5 minutes.
Ladle into bowls, add the crispy hellim cubes, sprinkle over a little more dried mint, pul biber and coarse black pepper, and a squeeze of lemon juice to taste.