Slice a strip lengthwise from the zucchini to expose the inside of the vegetable. Discard or reserve for another use. Cut the 2 ends from the zucchini to make straight edges. Cut the zucchini lengthwise into ½-inch strips.
Brush both sides of the zucchini pieces with olive oil. Season well kosher salt and freshly ground black pepper.
Lay the zucchini pieces on the grill. Cook until the zucchini is very tender, but not mushy, about 3 minutes per side, moving the pieces during cooking to ensure even browning.
Remove the zucchini from the grill and set on a wire cooling rack.
Spread about 1 teaspoon of goat cheese on each zucchini slice. Divide the red pepper and capers evenly amongst the zucchini.