Start cooking the pasta according to package directions in salted boiling water, as this takes the longest. While the pasta cooks, melt butter in a large skillet over medium heat. Cut the sourdough into 1-inch cubes, then add to the melted butter with a pinch of salt. Toss frequently until the croutons are golden brown and crisp on all sides, about 5-7 minutes. Remove from heat and let cool on a plate. I like to let my croutons cool completely so they stay crunchy and don't soften in the salad.
While the pasta finishes cooking, make the dressing by combining the minced garlic, yogurt, finely grated parmesan, mustard, vinegar, anchovy paste, lemon juice, and a pinch each of salt and black pepper in a blender or food processor. Blend until smooth, then slowly drizzle in the olive oil while the blender is running to emulsify the dressing and create a creamy, cohesive sauce. I prefer adding the oil slowly rather than all at once because it helps the dressing emulsify properly and reach that restaurant-quality creamy texture.
Drain the cooked pasta and rinse lightly with cold water to stop the cooking and cool it slightly. In a large serving bowl, combine the chopped romaine with a pinch of salt and black pepper, then add about three-quarters of the Caesar dressing from Step 2 and toss gently to coat. Add the cooled pasta from this step and the finely grated parmesan (the 3 tbsp portion), then toss everything together until well combined. The key is not to overdress at this point—you want the salad components to shine through, not be drowning in dressing.
Transfer the salad to a serving platter or individual bowls. Top with the shaved parmesan, cooled croutons from Step 1, red pepper flakes, and chopped fresh parsley for color and freshness. Serve immediately with the remaining Caesar dressing on the side so guests can add more to their preference.