Roast garlic by cutting the top of the bulb off, exposing the tops of the garlic cloves; place in a small oven safe dish, drizzle olive oil over garlic bulb and roast in oven at 375° for 30-40 minutes until garlic is fragrant, lightly browned, and soft. Allow to cool until easy to handle. To remove the roasted garlic from the bulb, squeeze the bulb at the bottom releasing the cloves out the cut end. Cloves should pop right out. Separate out 5-6 cloves for the aioli and save remaining cloves for another recipe, or freeze for later use.
If using an immersion blender, add roasted garlic and remaining ingredients to a tall cup or jar large enough at the base for the blender to fit with at least an inch of space around the blender. Immerse the blender in the jar, using the blender to gently lift and stir while processing on high until the aioli is well mixed and creamy. If using a food processor, add garlic and remaining ingredients except oil to food processor bowl and pulse to combine. Then, while the machine is running, slowly pour oil in a steady stream while processing until aioli is well mixed and creamy.
The aioli will thicken when refrigerated. Use within one week.
Although a food processor does a fine job, an immersion blender is ideal and foolproof. The ingredients can all go in together, eliminating the step "slowly add the oil" so the oil can emulsify. The garlic can be roasted ahead of time and refrigerated until ready to make the Aioli.