Preheat oven to 350 degrees F.
Roast the nuts in a shallow baking pan in middle of oven until golden, 10 to 15 minutes. Rub warm nuts in a kitchen towel to remove skins (don't worry if some skins don't come off). Cool the nuts and then coarsely chop.
Whisk together vinegar, lemon juice, shallot, sugar, salt, and pepper, to taste. Add oils in a slow stream, whisking until combined well.
Toss together arugula, strawberries, hazelnuts, and dressing. Divide salad among plates and dot with the goat cheese.