In a large heavy-bottomed skillet, heat 1/4 cup plus 1 tablespoon olive oil over medium heat. Add tomato paste and cook, stirring, about 1 minute. Add sugar and cook, stirring, until well combined. Add vinegar and cook, stirring, to deglaze pan.
Add tomato halves, thyme, bay leaves, salt, and remaining 3/4 cup plus 3 tablespoons olive oil. Reduce heat to medium-low and continue cooking until olive oil is bright red in color, 5 to 6 minutes.
Remove bay leaves and thyme and transfer marmalade to refrigerator until chilled. Marmalade can be stored in an airtight container, refrigerated, for up to 1 week.