For the soup: Heat the oil in a large pot over medium heat, just until the oil shimmers. Add the butter and wait until its foam almost subsides, then add the onion and 1/4 teaspoon of the salt. Cook for about 5 minutes, stirring occasionally, until the onion softens and turns light golden. Add the garlic, paprika or Aleppo pepper and the cayenne pepper, if desired; cook, stirring, for 1 minute. Add the tomato paste and tomatoes; adjust the heat as needed to maintain steady bubbles at the edges. Cook uncovered for about 3 minutes, stirring once or twice and scraping the bottom to release any browned bits, until the mixture thickens slightly.
Increase the heat to medium-high. Stir in the bulgur, lentils, broth, water and the remaining 1/4 teaspoon salt; bring to a boil, then reduce the heat to medium or medium-low to maintain gentle bubbling. Cover and cook for 25 to 40 minutes, until the soup thickens, with the bulgur retaining a slight bite. Season with 1 tablespoon of the lemon juice and the black pepper. Taste and add lemon juice (up to 1 more tablespoon) or adjust the seasoning as needed. Cover and let sit for a few minutes.
While the soup rests, make the minted butter: Heat the oil and butter in a small skillet over medium heat. Once the butter has melted, add the mint and cayenne or Aleppo pepper (which will sizzle) and stir for about 15 seconds until fragrant. Immediately remove from the heat; pour the minted olive oil butter into a small serving bowl.
Divide the soup among individual bowls. Drizzle a little of the minted butter on top of each portion. Serve right away.