Finely mince garlic cloves or smash them through a garlic press. Place on a cutting board.
Add kosher salt and smash together with the garlic with the flat of a chef's knife, scoop it into a pile, and smash it again until it forms a relatively smooth paste.
Dice the roasted red pepper and place in a paper towel or clean kitchen towel. Squeeze out any excess liquid (otherwise your aioli will be runny). You don't have to be gentle with it, since you'll be pureeing it anyway.
Add garlic paste, mayonnaise, lemon juice, olive oil, roasted red peppers, and cayenne pepper to an immersion blender cup or the bowl of a mini food processor. Blend until smooth.
Spread on a sandwich or enjoy as a dip! Store refrigerated in an airtight container for up to a week.