Preheat the oven to 400 degrees F. Toss strawberries with sugar and scatter (allowing plenty of room) on a parchment-lined rimmed baking sheet. Roast 15 - 20 minutes or until strawberries have released their juices and edges have just begun to caramelize. Scrape berries and any juice into a blender and puree. Set aside to cool.
Meanwhile, make ginger simple syrup. Heat water in a medium saucepan over medium-high heat. Stir in sugar, and heat just until sugar dissolves. Remove from heat, grate in ginger (no need to peel it), and set aside to steep 10 minutes. Use a fine mesh strainer to strain syrup into a jar.
To make lemonade, pour lemon juice into a large pitcher. Stir in simple syrup, roasted strawberry puree, and water. Add more simple syrup to taste (keep in mind that the ice will dilute the lemonade). Serve over ice with sliced strawberries and lemon slices.
Roasted strawberry lemonade and will keep well covered in the fridge for 3 days. Leftover strawberry syrup will keep in the fridge 3 days as well – consider adding it to yogurt or smoothies. Simple syrup will keep well in a sealed jar in the fridge for about a month.