Heat 1 tablespoon oil in a large skillet over medium-high heat until shimmering.
Add the pepper strips and sauté until lightly browned at the edges and beginning to soften, about 5 minutes.
Reduce the heat to medium low, stir in the minced garlic, cover and continue cooking until soft, about 8 minutes longer, stirring occasionally. Season with salt and pepper to taste; remove from heat and stir in the vinegar.
Grill the bread over a medium heat, turning once, until golden brown on both sides, 1 to 2 minutes.
Place the toast slices on a large platter, rub the peeled garlic over the tops, and brush with the remaining olive oil.
Place a few slices of mozzarella cheese on each toast. Cover with a portion of the sautéed peppers, the top with a sprinkling of basil.