Place first 5 ingredients in a food processor; pulse 3 to 4 times or until chopped.
Combine panko, eggs, and beef; divide and shape into 4 (1 1/2-inch-thick) patties. Heat oil in a large nonstick skillet over medium-high. Sprinkle patties with 1/2 teaspoon salt and 1/2 teaspoon pepper. Add to pan; cook 4 minutes on each side (patties will not be cooked through). Remove patties from pan (do not wipe out pan).
Add mushroom mixture to pan; cook 6 minutes. Add wine; cook 2 minutes or until liquid almost evaporates, scraping to loosen browned bits. Stir in ketchup, honey, and Worcestershire; cook 3 minutes or until slightly thickened. Reduce heat to low. Add patties; cook 6 minutes or until done, turning to coat.
Place 1 tablespoon water and potatoes in a medium microwave-safe bowl; cover with plastic wrap. Microwave at high 6 minutes or until tender. Add 1/4 teaspoon salt, remaining 1/4 teaspoon pepper, milk, and butter to potatoes; mash to desired consistency.
Cook broccoli according to package directions; sprinkle with remaining 1/4 teaspoon salt. Serve with patties, sauce, and mashed potatoes.
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