Step 1

In batches, pulse mushrooms in a food processor, scraping down the sides once or twice, until finely chopped.

Step 2

Heat 2 tablespoons oil in a large pot over medium-high heat. Add the mushrooms and cook, stirring occasionally, until browned and the liquid has evaporated, 5 to 7 minutes. Add 4 tablespoons vinegar, 2 teaspoons paprika and garlic powder. Cook, stirring, for 3 minutes. Scrape into a large bowl and spread into a thin layer. Let cool for 10 minutes.

Step 3

Meanwhile, combine the remaining 1 tablespoon vinegar, 1/8 teaspoon salt and sugar in a small bowl. Add cucumbers and toss to combine. Combine mayonnaise, tomato paste and the remaining 1/2 teaspoon paprika in a small bowl. Set the cucumbers and aioli aside.

Step 4

Preheat grill to medium-high.

Step 5

Add beef and the remaining 1/2 teaspoon salt to the mushrooms. Mix well. Form into 8 equal patties, about 4 inches in diameter. Toss peppers with the remaining 1 tablespoon oil in a medium bowl. Transfer to a grill basket.

Step 6

Oil the grill rack. Grill the burgers, flipping once, until an instant-read thermometer inserted in the thickest part registers 155 degrees F, 8 to 10 minutes total. Grill the peppers, stirring occasionally, until browned, 5 to 8 minutes total.

Step 7

Top the burgers with cheese and cook until melted, 1 to 2 minutes more. Serve the burgers on buns with greens, onion, tomato, the peppers, cucumbers and aioli.