1. Understanding the Problem:
- We need to find the radius of the largest spherical scoop of ice cream that can fit inside an oblique circular cone such that at least 50% of the scoop's volume lies inside the cone.
- The height and base radius of the cone are both 1.

2. Setting Up the Geometry:
- Place the cone in a coordinate system with the vertex at $(0, 0, 1)$ and the base center at $(0, 0, 0)$.
- The cone is oblique, meaning the vertex is directly above the perimeter of the base. This implies the slant height of the cone is $\sqrt{1^2 + 1^2} = \sqrt{2}$.

3. Transforming the Problem:
- The problem is equivalent to finding the largest hemisphere that fits inside the cone with its center on the base plane. This is because exactly 50% of the scoop should be contained inside the cone at the maximum radius, and the base plane should go through the center of the scoop.

4. Analyzing the Cross-Section:
- Consider the cross-section of the cone and the hemisphere in the $xz$-plane. The cross-section of the cone is a right triangle with legs of length 1 and 2 (since the vertex is directly above the perimeter of the base).
- The center of the hemisphere lies on the base plane at $(x, 0, 0)$.

5. Finding the Largest Semicircle:
- The problem reduces to finding the radius of the largest semicircle that fits inside the right triangle with legs 1 and 2, with the center of the semicircle on the leg of length 2.
- Let the center of the semicircle be at $(x, 0)$ on the leg of length 2. The radius $r$ of the semicircle must satisfy:
$$r = \min(x, \frac{2 - x}{\sqrt{5}})$$
- This is because the semicircle can either be tangent to the hypotenuse or the leg of length 1.

6. Maximizing the Radius:
- To find the maximum radius, set the two expressions for $r$ equal:
$$x = \frac{2 - x}{\sqrt{5}}$$
- Solving for $x$:
$$x\sqrt{5} = 2 - x$$
$$x(\sqrt{5} + 1) = 2$$
$$x = \frac{2}{\sqrt{5} + 1}$$
- Rationalizing the denominator:
$$x = \frac{2}{\sqrt{5} + 1} \cdot \frac{\sqrt{5} - 1}{\sqrt{5} - 1} = \frac{2(\sqrt{5} - 1)}{5 - 1} = \frac{2(\sqrt{5} - 1)}{4} = \frac{\sqrt{5} - 1}{2}$$

The final answer is $\boxed{\frac{\sqrt{5} - 1}{2}}$