Preheat the oven to 400 degrees F. Line a baking sheet with parchment paper. Microwave the bacon drippings for 8 to 10 seconds to melt, if necessary. Combine the tomatoes, onion, bacon drippings, lemon pepper, red pepper flakes, 1/2 teaspoon salt and 1/4 teaspoon pepper in a large bowl. Spread on the prepared baking sheet and roast until slightly softened, about 10 minutes. Let cool. Reduce the oven temperature to 375 degrees F.
Coat the bottom and side of an 8- or 9-inch cast-iron skillet with cooking spray. Toss the tomato mixture with the lemon juice and asiago; spread evenly in the skillet. Place the puff pastry on top; trim the overhanging pastry with kitchen shears.
Transfer the skillet to the oven and bake until the pastry is dark golden, about 50 minutes. Let cool about 5 minutes, then place a large plate or platter on top of the skillet and invert the tart. Sprinkle with basil and let cool, about 5 more minutes. Cut into wedges.