Make the tomato jam. Hull the tomatoes, then rough chop and transfer them to a bowl along with any of the juices that accumulate on the cutting board; set aside.
In a medium-sized nonstick skillet, warm the oil over medium heat.
Add the shallot and garlic and cook, stirring constantly, until softened, for 1 ½ to 2 minutes.
Add the tomatoes and any juices, the sugar, vinegar and a generous pinch of salt, stir to combine and bring to a boil.
Reduce to a simmer and cook, stirring often and using the back of a wooden spoon to break up the tomatoes, until jammy, about 40 minutes. As the jam thickens, you may need to stir it more frequently.
Season with additional salt, as needed.
Transfer to a heat-resistant container and cool completely.
Make the basil oil. Using a high-powered blender, blend the basil and grapeseed oil on high until the basil is as pureed as possible.
Add the olive oil and blend on high until well combined.
Strain the mixture using a fine-mesh strainer and transfer to a storage container. Store in the refrigerator for up to 1 week, allowing the oil to come to room temperature before using. Shake before using.
To serve, swipe a very generous amount of the yogurt across a rimmed salad plate.
Top generously with tomato jam.
Scatter some dollops of the Calabrian chili paste over the jam.
Drizzle some basil oil over the top.
Sprinkle some toasted pine nuts evenly over the top.
Season with a sprinkling of flaky sea salt and garnish with basil leaves.
Serve with warm pita or focaccia.