Preheat oven to 375 degrees F.
Line a shallow baking dish with parchment paper, and place the 8 pepper halves on top, skin side down.
Top the peppers with anchovies and garlic.
Measure 1/4 cup of the torn basil leaves, and scatter them on top of the peppers. Season with salt and pepper and drizzle with 2 T olive oil.
Bake peppers for 35 to 40 minutes until they are tender and slightly charred around the edges. Remove from oven when done, and set aside to cool.
Place remaining 3/4 cup of basil into a blender with the remaining 1/3 cup of extra virgin olive oil and blend.
Season to taste with salt and pepper.
Arrange peppers on a serving platter. Top with torn Burrata cheese, tomato halves, olive halves, capers, a sprinkling of lemon zest and torn basil leaves, and a drizzle of basil oil. Finish off with salt and freshly ground pepper.