Preheat oven to 350°F. Combine shortbread cookies and 1/2 cup almonds in a food processor. Process until finely ground. Transfer crumbs to a 9-inch springform pan and then pat and press on bottom and 1 1/4 inches up side of pan. Bake for 10 minutes, until golden and fragrant. Let crust cool. Put remaining 1/4 cup almonds in a small baking pan and roast in oven until toasted, 8 to 10 minutes. Let cool and reserve for garnish.
Slice enough strawberries to measure 1 1/2 cups, and put in a medium saucepan with 1/3 cup sugar. Reserve remaining berries for garnish. Cook over medium heat, simmering until soft, about 6 minutes. In a large saucepan, heat 1 inch of water to a simmer. Beat together eggs and remaining 1/3 cup plus 2 1/2 Tbsp. sugar with a handheld electric mixer in a large bowl. Beat hot strawberry mixture into egg mixture. Place bowl on top of saucepan with simmering water. Beat mixture at medium-high speed until temperature reaches 140°F on an instant-read thermometer. Turn off heat and continue beating for 3 minutes. Remove bowl from saucepan and let strawberry custard cool. Cover mixture and chill for 30 minutes.
In a separate bowl, gradually beat heavy cream into mascarpone; beat until mixture is just stiff but not dry. Fold cream mixture into strawberry custard and then pour into crumb crust, smoothing top. Wrap well in plastic and freeze for at least 3 hours and up to 3 days.
To serve, unwrap and unmold cake; place on a round cake platter and garnish with toasted almonds and sliced strawberries.
Unmold ice cream easily. Before removing an ice cream mold from its pan, let it sit for 10 minutes. It'll slide right out.
Keep it fresh. Icebox cakes can be stored for up to three days. Ice cream cakes keep up to a week in the freezer, as long as they're wrapped well.
Get an instant-read thermometer. These handy gadgets look like large knitting needles with a temperature gauge on the end. They start at around $10; look for one with a large, easy-to-read dial.