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1
To make the aioli, preheat the oven to 375 degrees F.
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2
Place the garlic and tomato on a rimmed baking sheet, toss with the oil, and season with salt and pepper.
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3
Roast in the oven until the tomato is soft and the garlic is light golden brown, about 15 minutes.
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4
Slice the tomato in half crosswise and remove the seeds.
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5
Set aside to cool.
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6
Combine the garlic, tomato, lemon zest, lemon juice, thyme, and mayonnaise in a food processor and process until smooth; season with salt and pepper.
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7
Cover and refrigerate for at least 30 minutes to allow the flavors to meld.
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8
The aioli can be made up to 8 hours in advance and stored in an airtight container in the refrigerator.
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9
Cut the tuna into large pieces and then coarsely chop in a food processor.
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10
Do not overprocess.
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11
(Alternatively you can chop it by hand with a sharp knife.)
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12
Whisk together 2 tablespoons of the oil, the mayonnaise, mustard, anchovy paste, vinegar, olives, shallot, and capers in a large bowl.
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13
Add the tuna and basil and gently fold to combine.
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14
Divide the tuna mixture into 4 equal portions (about 6 ounces each).
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15
Form each portion loosely into a 3/4-inch-thick burger and make a deep depression in the center with your thumb.
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16
Place on a plate, cover with plastic wrap, and let chill in the refrigerator for at least 30 minutes before cooking.
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17
To cook the burgers, heat the remaining 2 tablespoons oil in a saute pan or griddle (nonstick or cast iron) until it begins to shimmer.
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18
Season both sides of each burger with salt and pepper.
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19
Cook the burgers until golden brown on the bottom sides, about 3 minutes.
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20
Turn over and continue cooking until medium, about 3 minutes longer.
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21
Spread the aioli on both sides of the rolls and place the burgers and mesclun greens on the bottom halves.
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22
Cover with the roll tops and serve immediately.
