Chop the garlic cloves, then sprinkle with some salt and smear to a paste, using the side of your knife blade. Transfer the garlic paste to a bowl and stir in the vinegar. Slowly whisk in the olive oil until emulsified. Fold in the anchovies, basil, olives, capers and pepper flakes. Taste and season with salt and pepper. Let sit at room temperature while you grill the vegetables.
Heat a grill to high. Brush the red and yellow peppers with oil and season with salt and pepper. Grill, covered, until charred on all sides and soft, 3 to 4 minutes per side. Remove the peppers to a bowl, cover tightly with plastic wrap and let steam for 15 minutes. Meanwhile, put the tomatoes in a grill basket and grill, covered, tossing occasionally, until charred in spots and softened, about 5 minutes. Set aside.
Brush the portobellos with canola oil and season on both sides with salt and pepper. Grill on both sides, covered, until a paring knife inserted in the center meets no resistance, 3 to 5 minutes per side. While the portobellos are cooking, peel, seed and thinly slice the peppers.
Turn the portobellos gill-side up and top each with some of the goat cheese. Grill, covered, until the cheese softens, about 2 minutes. Transfer to a platter and top each mushroom with some tomatoes and peppers. Spoon a generous amount of dressing over each "pizza" and serve warm.