In a large bowl, combine all-purpose flour, instant yeast, sugar, and sea salt. Add olive oil and lukewarm water. Mix together with a wooden spoon or fork (different flours absorb liquid differently, so if it's too dry, add a little bit of water), then transfer to a lightly floured surface and knead with your hands for about 5-10 minutes until the dough is soft.
Place the dough into a lightly oiled bowl, cover with a kitchen towel, and set aside to rise in a warm place for about 1 hour until doubled in size (it can take longer in a cooler place).
Grease a 9x5 inch (25x13cm) loaf pan with oil or vegan butter. Set aside.
After the dough has doubled, punch the dough down and place it on a lightly floured surface. Sprinkle a little extra flour on top of the dough. Use a rolling pin to roll out the dough into a large rectangle, about 16in x 10in (40cm x 25cm). Spread tomato sauce all over the dough, almost reaching the edges. Sprinkle Italian herbs, vegan parmesan, and vegan mozzarella over the tomato sauce. Roll the dough lengthwise into a tight log.
Using a sharp knife, cut the log in half lengthwise. Twist the halves around each other about four times, and pinch together both ends. Place the braid into the greased loaf pan, cover with a kitchen towel, and let rise in a warm place for 30-60 minutes until doubled in size.
Preheat the oven to 355°F (180°C). After the babka has doubled in size, place it into the oven and bake for about 45 minutes. If the babka starts to get too brown on the top during the baking, cover the top with aluminum foil. Remove from the oven and let cool for 15 minutes in the pan. Then remove the babka from the pan and place it on a cooling rack to cool completely before cutting into it.