1. Preheat the oven to 450 degrees.
2. Cut off the top of the garlic bulb to expose the cloves. Place on a square piece of tin foil, drizzle with olive oil, and sprinkle with salt and pepper. Tightly wrap the foil around the head of garlic and place in the oven for 30 minutes, or until the cloves are golden brown and starting to burst from their skins.
3. Remove from the oven when the garlic is finished cooking and let cool completely.
4. While the garlic is cooking and cooling, fill a large sauce pot halfway full with water and bring to a boil. When the water is boiling, slowly drop the eggs in the water using a slotted spoon. Cover and let cook for 6 minutes and 30 seconds.
5. After the time has passed, remove the eggs from the water and transfer to an ice bath to cool completely.
6. While the eggs are chilling, place the bell peppers over an open flame (either a grill or gas stove top) and cook until all of the skin is blackened. Transfer the charred peppers, still hot, to a plastic bag to steam for a couple of minutes. *The steam will help remove the blackened skin and give them time to cool a bit.
7. When the peppers are cool enough to handle, remove the top core and seeds. Peel off the blackened skin and transfer to a food processor. Squeeze the garlic cloves into the food processor and combine with the almonds and cayenne pepper.
8. Blend everything together while slowly pouring the olive oil through the spout. Add enough olive oil until the romesco reaches your desired thickness. **If you want it to be looser and used as a dressing or marinade, add all of the olive oil. If you want it to be thicker and used as a topping or a sauce, add a little less olive oil.
9. Squeeze in the juice of half the lime and season to taste with salt and pepper. Transfer to an airtight container and store in the refrigerator until you are ready to serve.
10. When you are ready to serve, remove the eggs from the ice water and peel. Toss the cilantro leaves with some lime juice. Slice the eggs in half and place on a platter. Top with a spoonful of romesco sauce and garnish with the lime-dressed cilantro leaves.