Wash and dry apricots. Cut in half (removing stones and stems). Brush cut sides with olive oil.
Place cut side down on hot grill surface. Cook until edges just begin to bubble and slight caramelization occurs. Flip and cook uncut surfaces until fruit edges are softened.
Cut grilled apricots into ½” chunks and set aside to cool.
Whisk egg white in bowl over hot water. When thoroughly aerated, slowly add sugars in a continuous stream while continuing to whisk. Continue whisking whites and sugars until hot to touch (~120F). Remove from heat and beat until soft peaks form.
Allow to cool further. Beat in cream 2 tbsp at a time for the first cup, then ¼ cup at a time. Don’t worry about the mixture being very fluid, you are not trying to make buttercream here. Cover and chill.
Add cooled cream mix and apricots to ice cream maker and process.
Remove from ice cream maker and transfer to sealed containers and freeze until firm.
Best if allowed to sit out of the freezer for 5 minutes before serving.