Whisk the olive oil, mayo, vinegar, Dijon, shallot, lemon juice and some salt and pepper together in a small bowl.
Turn a gas burner on medium-high and cook the whole red pepper directly on the flame, turning occasionally, until charred all over, about 10 minutes. Put the pepper in a small bowl and cover with plastic wrap or a dish towel. Let stand for 15 minutes to steam, then use the back of a paring knife to remove the skin from the pepper. Remove the stem and seeds, then cut the pepper into thin strips.
Toss the arugula and mint together in a wide shallow bowl or platter, then arrange the mozzarella and peppers on top. Drizzle with the vinaigrette and serve.